  Ideal crystalline membranes, realized by graphene and other atomic
monolayers, exhibit rich physics - a universal anomalous elasticity of the
critical "flat" phase characterized by a negative Poisson ratio, universally
singular elastic moduli, order-from-disorder and a crumpling transition. We
formulate a generalized $D$-dimensional field theory, parameterized by an
$O(d)\times O(D)$ tensor field with an {\it energetic} longitudinal constraint.
For a soft constraint the resulting field theory describes a new class of a
fluctuating "tattered" membranes, exhibiting a nonzero density of topological
connectivity defects - slits, cracks and faults at an effective medium level.
For hard, infinite-coupling constraint, the model reproduces the conventional
crystalline membrane and its crumpling transition, and thereby demonstrates the
essence of the difference between an elastic membrane and conventional field
theories. Two additional fixed points emerge within the critical manifold, (i)
globally attractive, "isotropic" $O(d)\times O(D)$, and (ii) "transverse",
which in $D=2$ is the exact "dual" of the elastic membrane. Their properties
are obtained in general $D,d$ from the renormalization group and the
self-consistent screening analyses.
