• fluid membranes
• quantum gravity
• string
• nonlinear realization

Effective field theories for membranes have found several applications in physics. They are
used in condensed matter physics and biological physics and have also found, in the last few
decades, space in string theory and gravitational theories. Studying such theories can thus
be useful to develop common languages and tools that connect various aspects of the physics
world. Examples of research objects of interest are the study of graphene, which mechanically
can be described by a crystalline membrane, and cells’ membranes that can be described by
a fluid membrane model. The model that we have chosen to study in detail in this thesis is
related to the Helfrich action for membranes, that was originally used to describe cells’ lipid
bilayers.
The phase-diagram of a general membrane can include many kinds of phases, which are
separated by phase-transitions. However, in general, we do not know what the order of these
phase-transition is. Therefore, in the thesis we concentrated our study on the mechanical
properties of a general fluid N -dimensional membrane embedded in a D-dimensional bulk
space. We searched especially for second order phase-transition because we do know that in
this case, we find universal exponents, that are in common with all membranes of the same
family independently of their microscopic origin.
Membrane theories can be derived from pure group theory analysis. The presence of the
membrane, whatever type it is, does spontaneously break the group of isometries of the bulk
space. The broken generators do correspond to infinitesimal transformations that mix mem-
brane elements with bulk elements and can be related to quantities describing the extrinsic
properties of the membrane’s geometry. Of course, this is the case of fluid membranes in which
any translation tangent to it is still locally invariant, while for other types of membrane, like
the crystalline one, this does not hold.
A fluid membrane theory, that geometrically is constructed up to terms proportional to the
square of the extrinsic curvatures and the Ricci scalar, generally presents two distinct types of
mechanical phase states. The membrane can be either crumpled or in an extended phase. If
the transition is a second order one, then there is a critical temperature that separates these
two possible phases and, in particular, we have that for temperatures smaller than the critical
one, the membrane is in the extended phase. This separation was studied in our case by using
the Wilsonian renormalization group method, with the use of momentum shell regularization.
Small thermal fluctuations of the fluid membrane, which is represented by a differentiable
manifold with no boundaries, do give rise to a phase-diagram and the state of the membrane
depends on the bending coupling constants associated to extrinsic curvatures (the bending
rigidity) and on the dimensions of both the bulk and the membrane. For technical reasons it
is also convenient to try to set the area constant in the partition function. This would give us
statistical observables, that directly depend on it.
1We argue that a fluid membrane model is of particular importance in some given limits to
obtain gravitational or gravity-like theories. The extended Helfrich model with no curvature
terms reduces to the Nambu-Goto action of string theory. Such action is also strictly related to
the Polyakov action, which can be interpreted as a theory of two-dimensional quantum gravity
coupled to multiple scalar matter fields. Thus, one can think that, in this case, the limit
of embedding dimension going to zero, D → 0, can be viewed as a matterless limit for the
Polyakov action, implying that the same limit in our effective membrane model may be related
to a new model of pure two-dimensional quantum gravity. In fact, by doing so we have obtained
indications of a theory that accounts only for the ghost sector of quantum gravity. By extending
the previous analogy to higher dimensions, we obtain results that agree with the asymptotic
safety conjecture of quantum gravity. Unfortunately, even if we take a small expansion for the
membrane dimension, N ' 2, to higher dimensions, we do not obtain an expected graviton
contribution. Such peculiar behavior could come from the intrinsic properties of our model or
from our choice of regularization.
Another important limit, that we have considered, is D → N . We believe that such limit
corresponds to a theory of dynamical diffeomorphisms, that may find applications in cosmol-
ogy. In particular, the action constructed with only the lowest order trace in derivatives of the
diffeomorphisms, corresponds to the usual action for nonlinear sigma model and has the same
form of the Polyakov action with a different interpretation of the fields used to build it. By
applying the said limit when we build our effective membrane model, we have that the diffeo-
morphisms’ equations of motions are the same. In fact, we can relate our embedding functions
to reparameterization of the membrane with a general target space metric.
In the conclusions we tried to give ideas for a sounder geometrical interpretation of the
limits, by searching for a bulk space, that admits the correct isometries, that are spontaneously
broken by the presence of the membrane and reproduce the two limits. We hope that such a
new model gives back the same results, but with a clearer physical interpretation.