Use of Quadratic Differentials for Description of Defects and Textures in Liquid Crystals and 2+1 Gravity

TL;DR

This paper explores the application of quadratic differentials to describe defects in liquid crystals and the structure of 2+1 gravity, revealing new results including discrete mass spectra and connections to hyperbolic 3-manifolds.

Contribution

It extends the theory of quadratic differentials to 2+1 gravity, removing mass restrictions and linking the dynamics to hyperbolic 3-manifolds and knot theory.

Findings

Masses in 2+1 gravity are discrete and sum rules restrict possible values.

The dynamics relate to hyperbolic 3-manifolds and knot complements.

New results extend previous theories and remove mass restrictions.

Abstract

The theory of measured foliations which is discussed in PartI(hep-th/9901040) in connection with train tracks and meanders is shown to be related to the theory of Jenkins-Strebel quadratic differentials by Hubbard and Masur (Acta Math.Vol.142,221(1979)). Use of quadratic differentials not only provides an adequate description of defects and textures in liquid crystals but also is ideally suited for study of 2+1 classical gravity which was initiated in the seminal paper by Deser, Jackiw and 't Hooft (Ann.Phys.Vol.152,220(1984)). In this paper not only their results are reproduced but, in addition, many new results are obtained. In particular, using the results of Rivin (Ann.Math.Vol.139,553(1994)) the restriction on the total mass of the 2+1 Universe is removed. It is shown that the masses can have only discrete values and, moreover, the theoretically obtained sum rules forbid the existence of some of these values. The dynamics of 2+1 gravity which is associated with the dynamics of train tracks is being reinterpreted in terms of the emerging hyperbolic 3-manifolds. The existence of knots and links associated with complements of these 3-manifolds is highly nontrivial and requires careful proofs. The paper provides a concise introduction into this topic. A brief discussion of connections with related physical problems, e.g.string theory, classical and quantum billiards, dynamics of fracture, protein folding, etc. is also provided.