Vielbein with mixed dimensions and gravitational global monopole in the planar phase of superfluid 3He
G. E. Volovik
Low Temperature Laboratory, Aalto University, School of Science and Technology, P.O. Box 15100, FI-00076 AALTO, Finland
Landau Institute for Theoretical Physics, 142432 Chernogolovka, Russia
The planar phase of superfluid 3He has Dirac points in momentum space and the
analog of Dirac monopole in the real space.
Here we discuss the combined effect of Dirac point and Dirac monopole. It is
shown that in the presence of the monopole the effective metric acting on Dirac
fermions corresponds to the metric produced by the global monopole in general
relativity: it is the conical metric.
Another consequence is that the primary variable, which gives rise to the
effective metric, is the unusual vielbein field in the form of the 4× 5
matrix, as distinct from the conventional 4× 4 matrix of the tetrad field
in tetrad gravity.