M-theory circle duality ($R_{11}=g_s{\ell }_s,{\ell }_p^3=g_s{\ell }_s^3$)

Compactifying M-theory on a circle gives type IIA superstring theory. The circle radius sets its coupling. Multiplying M2-brane and M5-brane tensions by the circle circumference gives the fundamental string and D4-brane tensions, while unwrapped objects give D2-branes and NS5-branes. Momentum and a Kaluza-Klein monopole supply D0-brane and D6-brane charges. The circle dictionary and worldvolume reductions are developed in Four Lectures on M-theory.

D0-branes are circle momentum states, D2-branes are unwrapped M2-branes, D4-branes are circle-wrapped M5-branes, and D6-branes arise from Kaluza-Klein monopoles with the M-circle as fibre. Wrapped M2-branes give fundamental strings, and unwrapped M5-branes give NS5-branes. The D8-brane is a source for Romans mass and is outside this ordinary massless-circle dictionary. Tension matching uses $R_{11}=g_s{\ell }_s$ and ${\ell }_p^3=g_s{\ell }_s^3$.