  In this work we consider an interacting quantum field theory on a curved
two-dimensional manifold that we construct by geometrically deforming a flat
hexagonal lattice by the insertion of a defect. Depending on how the
deformation is done, the resulting geometry acquires a locally non-vanishing
curvature that can be either positive or negative. Fields propagating on this
background are forced to satisfy boundary conditions modulated by the geometry
and that can be assimilated by a non-dynamical gauge field. We present an
explicit example where curvature and boundary conditions compete in altering
the way symmetry breaking takes place, resulting in a surprising behaviour of
the order parameter in the vicinity of the defect. The effect described here is
expected to be generic and of relevance in a variety of situations.
