From Ginzburg–Landau to Hilbert–Einstein via Yamabe

In this paper, based on some mathematical results obtained by Yamabe, Osgood, Phillips and Sarnak, we demonstrate that in dimensions three and higher the famous Ginzburg–Landau equations used in the theory of phase transitions can be obtained (without any approximations) by minimization of the Riemannian-type Hilbert–Einstein action functional for pure gravity in the presence of cosmological term. We use this observation in order to bring to completion the work by Lifshitz (done in 1941) on group-theoretical refinements of the Landau theory of phase transitions. In addition, this observation allows us to develop a systematic extension to higher dimensions of known string-theoretic path integral methods developed for calculation of observables in two-dimensional conformal field theories.