Curved space and amorphous structures Part II. Gauge theories

Abstract

In Part I we discussed how amorphous structures can be regarded as mappings from curved spaces, and how such mappings provided us a perspective on the line defects of topological origin that possibly exist in amorphous materials. The existence of such defects also permits us to speculate that the liquid, crystalline and amorphous states differ largely in the spatial organization and the kinetics of such defects. Gauge theories offer a means of quantifying such notions, and Part II reviews recent attempts in this direction. We commence with some mathematical preliminaries concerning differential geometry, and then explain what is meant by a gauge theory by considering a few examples from condensed matter physics. We then review the recent works of Rivier and Duffy, of Nelson and co-workers and of Sethna concerning a possible gauge/gauge-type theory for glass. We conclude by making a broad assessment of the progress thus far and of the territory that lies beyond.