  In underdoped cuprates the normal state is highly anomalous and is
characterized as a pseudogap phase. The question of how to describe the
``normal'' core 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
normal state as well as the vortex core. Interestingly, the pseudogap persists
inside the core. We also found that it is likely that the core consists of a
state which breaks translational symmetry due to the existence of staggered
current which generate staggered magnetic field with very slow dynamics. This
staggered flux state is likely to be the ground state for magnetic fields
higher than $H_{c2}$. Experiments to test this picture are proposed.
