From elasticity tetrads to rectangular vielbein

TL;DR

This paper explores the concept of elasticity vielbein in condensed matter physics, extending it to rectangular vielbein structures that could connect to gauge fields in the Standard Model and GUT, with implications for gravity theories.

Contribution

It introduces the idea of rectangular vielbein fields in condensed matter systems and discusses their potential role in extending Einstein-Cartan gravity to include higher symmetry groups.

Findings

Rectangular vielbein can emerge near Dirac points in materials.

These vielbeins relate to phase fields and gauge fields in condensed matter.

Potential extension of gravity theories with higher group structures.

Abstract

The paper is devoted to the memory of Igor E. Dzyaloshinsky. In our common paper I.E. Dzyaloshinskii and G.E. Volovick, Poisson brackets in condensed matter, Ann. Phys. {\bf 125} 67--97 (1980), we discussed the elasticity theory described in terms of the gravitational field variables -- the elasticity vielbein $E_\mu^a$. They come from the phase fields, which describe the deformations of crystal. The important property of the elasticity vielbein $E^a_\mu$ is that in general they are not the square mstrices. While the spacetime index $\mu$ takes the values $\mu=(0,1,2,3)$, in crystals the index $a=(1,2,3)$, in vortex lattices $a=(1,2)$, and in smectic liquid crystals there is only one phase field, $a=1$. These phase fields can be considered as the spin gauge fields, which are similar to the gauge fields in Standard Model (SM) or in Grand Unification (GUT). On the other hand, the rectangular vielbein $e^\mu_a$ may emerge in the vicinity of Dirac points in Dirac materials. In particular, in the planar phase of the spin-triplet superfluid $^3$He the spacetime index $\mu=(0,1,2,3)$, while the spin index $a$ takes values $a=(0,1,2,3,4)$. Although these $(4 \times 5)$ vielbein describing the Dirac fermions are rectangular, the effective metric $g^{\mu\nu}$of Dirac quasiparticles remains (3+1)-dimensional. All this suggests the possible extension of the Einstein-Cartan gravity by introducing the rectangular vielbein, where the spin fields belong to the higher groups, which may include SM or even GUT groups.