  We discuss ground state selection by quantum fluctuations in frustrated
magnets in a strong magnetic field. We show that there exist dynamical
symmetries -- one a generalisation of Henley's gauge-like symmetry for
collinear spins, the other the quantum relict of non-collinear weathervane
modes -- which ensure a partial survival of the classical degeneracies. We
illustrate these for the case of the kagome magnet, where we find zero-point
energy differences to be rather small everywhere except near the collinear
`up-up-down` configurations, where there is rotational but not translational
symmetry breaking. In the effective Hamiltonian, we demonstrate the presence of
a term sensitive to a topological `flux'. We discuss the connection of such
problems to gauge theories by casting the frustrated lattices as medial
lattices of appropriately chosen simplex lattices, and in particular we show
how the magnetic field can be used to tune the physical sector of the resulting
gauge theories.
