The statistical mechanics of membranes

The fluctuations of two-dimensional extended objects (membranes) is a rich and exciting field with many solid results and a wide range of open issues. We review the distinct universality classes of membranes, determined by th e local order, and the associated phase diagrams. After a discussion of sev eral physical examples of membranes we turn to the physics of crystalline ( or polymerized) membranes in which the individual monomers are rigidly boun d. We discuss the phase diagram with particular attention to the dependence on the degree of self-avoidance and anisotropy. In each case we review and discuss analytic, numerical and experimental predictions of critical expon ents and other h-ey observables. Particular emphasis is given to the result s obtained from the renormalization group epsilon -expansion. The resulting renormalization group Bows and fixed points are illustrated graphically. T he full technical details necessary to perform actual calculations are pres ented in the Appendices, We then turn to a discussion of the role of topolo gical defects whose liberation leads to the hesatic and fluid universality classes, We finish with conclusions and a discussion of promising open dire ctions for the future, (C) 2001 Elsevier Science B,V, All lights reserved.