  We consider the low-energy electronic properties of graphene cones in the
presence of a global Fries-Kekul\'e Peierls distortion. Such cones occur in
fullerenes as the geometric response to the disclination associated with
pentagon rings. It is well known that the long-range effect of the disclination
deficit-angle can be modelled in the continuum Dirac-equation approximation by
a spin connection and a non-abelian gauge field. We show here that to
understand the bound states localized in the vicinity of a pair of pentagons
one must, in addition to the long-range topological effects of the curvature
and gauge flux, consider the effect the short-range lattice disruption near the
defect. In particular, the radial Dirac equation for the lowest
angular-momentum channel sees the defect as a singular endpoint at the origin,
and the resulting operator possesses deficiency indices $(2,2)$. The radial
equation therefore admits a four-parameter set of self-adjoint boundary
conditions. The values of the four parameters depend on how the pentagons are
distributed and determine whether or not there are zero modes or other bound
states.
