In underdoped cuprates the normal state is highly anomalous and is characte
rized as a pseudogap phase. The question of how to describe the "normal" co
re of a superconducting vortex is an outstanding open problem. We show that
the SU(2) formulation of the t-J model provides a description of the norma
l state as well as the vortex core. Interestingly, the pseudogap persists i
nside the core. We also found that it is likely that the core consists of a
state that breaks translational symmetry due to the existence of a stagger
ed current that generates a staggered magnetic field with very slow dynamic
s. This staggered-flux stare is likely to be the ground state for magnetic
fields higher than H-c2. Experiments to test this picture are proposed.