Measure Factors, Tension, and Correlations of Fluid Membranes

TL;DR

This paper derives geometrical correction factors for statistical ensembles of fluid membranes, ensuring coordinate-invariance and consistency in free energy and correlation calculations, with implications for membrane physics and string theory.

Contribution

The authors introduce a simple geometrical derivation of correction factors for membrane ensemble measures, clarifying the role of Faddeev-Popov and Liouville-like terms.

Findings

Derived explicit formulas for effective frame tension.

Calculated two-point correlation functions.

Confirmed coordinate-invariance at lowest order.

Abstract

We study two geometrical factors needed for the correct construction of statistical ensembles of surfaces. Such ensembles appear in the study of fluid bilayer membranes, though our results are more generally applicable. The naive functional measure over height fluctuations must be corrected by these factors in order to give correct, self-consistent formulas for the free energy and correlation functions of the height. While one of these corrections -- the Faddeev-Popov determinant -- has been studied extensively, our derivation proceeds from very simple geometrical ideas, which we hope removes some of its mystery. The other factor is similar to the Liouville correction in string theory. Since our formulas differ from those of previous authors, we include some explicit calculations of the effective frame tension and two-point function to show that our version indeed secures coordinate-invariance and consistency to lowest nontrivial order in a temperature expansion.