From elasticity tetrads to rectangular vielbein
Abstract
The paper is devoted to the memory of Igor E. Dzyaloshinsky. In our common paper Dzyaloshinskii and Volovick (1980), we discussed the elasticity theory described in terms of the gravitational field variables  the elasticity vielbein E_{μ}^{a} . They come from the phase fields, which describe the deformations of crystal planes. The important property of the elasticity vielbein E_{μ}^{a} is that in general they are not the square matrices. While the spacetime index μ takes the values μ =(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_{a}^{μ} may emerge in the vicinity of Dirac points in Dirac materials. In particular, in the planar phase of the spintriplet superfluid ^{3}He the spacetime index μ =(0 , 1 , 2 , 3) , while the spin index a takes values a =(0 , 1 , 2 , 3 , 4) . Although these (4 × 5) vielbein describing the Dirac fermions are rectangular, the effective metric g^{μν} of Dirac quasiparticles remains (3+1)dimensional. All this suggests the possible extension of the EinsteinCartan gravity by introducing the rectangular vielbein, where the spin fields belong to the higher groups, which may include SM or even GUT groups.
 Publication:

Annals of Physics
 Pub Date:
 December 2022
 DOI:
 10.1016/j.aop.2022.168998
 arXiv:
 arXiv:2205.15222
 Bibcode:
 2022AnPhy.44768998V
 Keywords:

 Tetrad;
 Vielbein;
 Elasticity theory;
 EinsteinCartan gravity;
 Dirac fermions;
 Superfluid <SUP>3</SUP>He;
 Physics  Classical Physics;
 Condensed Matter  Other Condensed Matter;
 General Relativity and Quantum Cosmology;
 High Energy Physics  Phenomenology
 EPrint:
 6 pages, no figures, accepted for the issue of Ann. Phys. devoted to memory of I.E. Dzyaloshinsky