  We study the spin-1/2 two-leg Heisenberg ladder with four-spin ring exchanges
under a magnetic field. We introduce an exact duality transformation which is
an extension of the spin-chirality duality developed previously and yields a
new self-dual surface in the parameter space. We then determine the magnetic
phase diagram using the numerical approaches of the density-matrix
renormalization-group and exact diagonalization methods. We demonstrate the
appearance of a magnetization plateau and the Tomonaga-Luttinger liquid with
dominant vector-chirality quasi-long-range order for a wide parameter regime of
strong ring exchange. A "nematic" phase, in which magnons form bound pairs and
the magnon-pairing correlation functions dominate, is also identified.
