The Effective Field Theory approach towards membrane-mediated interactions between particles.

Fluid lipid membranes can mediate forces between particles bound to them: A local deformation of the surface geometry created by some object spreads to distant regions, where other objects can respond to it. The physical characteristics of these geometric interactions, and how they are affected by thermal fluctuations, are well described by the simple continuum curvature-elastic Hamiltonian proposed 40 years ago by Wolfgang Helfrich. Unfortunately, while the underlying principles are conceptually straightforward, the corresponding calculations are not-largely because one must enforce boundary conditions for finite-sized objects. This challenge has inspired several heuristic approaches for expressing the problem in a point particle language. While streamlining the calculations of leading order results and enabling predictions for higher order corrections, the ad hoc nature of the reformulation leaves its domain of validity unclear. In contrast, the framework of Effective Field Theory (EFT) provides a systematic way to construct a completely equivalent point particle description. In this review we present a detailed account for how this is accomplished. In particular, we use a familiar example from electrostatics as an analogy to motivate the key steps needed to construct an EFT, most notably capturing finite size information in point-like "polarizabilities," and determining their value through a suitable "matching procedure." The interaction (free) energy then emerges as a systematic cumulant expansion, for which powerful diagrammatic techniques exist, which we also briefly revisit. We then apply this formalism to derive series expansions for interactions between flat and curved particle pairs, multibody interactions, as well as corrections to all these interactions due to thermal fluctuations.