From elasticity tetrads to rectangular vielbein

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 spin-triplet superfluid ³He 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 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.