  We compute the structure of flux $h/(2e)$ vortices in a $d$-wave
superconductor using a continuum SU(2) gauge theory of 2 flavors of charge $e$,
SU(2)-fundamental Higgs bosons. Period-2 charge order is present near the
vortex center. Upon coupling the electrons to the superconducting and charge
order parameters, we find that the electronic local density of states does not
have a zero-bias peak, in contrast to BCS theory. But there are sub-gap peaks
at positive and negative bias, and these exhibit anti-phase periodic spatial
modulations, as observed in scanning tunneling microscopy experiments in the
underdoped cuprates (K. Matsuba et al., J. Phys. Soc. Jpn. 76, 063704 (2007)).
