We propose a complete, $d=6\mathrm{}$ covariant and kappa-symmetric, action for the $M$ theory five-brane propagating in $D=11\mathrm{}$ supergravity background. This opens a direct way of relating a wide class of super- $p$-brane solutions of string theory with the five-brane of $M$ theory, which should be useful for studying corresponding dualities and nonperturbative aspects of these theories.

We observe that the N3 entropy behavior of near-extremal M5 branes can be reproduced from SYM side with the role of Myersʼ terms. We start by generalizing the Klebanov–Tseytlin (KT) supergravity solution that displays the N3 entropy behavior. The new feature of the general solution is visibility of the “internal” degrees of the M5 branes, i.e., the M0 branes and the M2 branes. With the rationale provided by the supergravity analysis, we consider a D0 brane quantum mechanical setup with Myersʼ terms. Using localization technique, we show that the leading N3 behavior of the free energy comes from the “classical contribution” with the rest subleading