I work in holography and I'm trying to get my feet when in non-relativistic holography. Can someone explain exactly what an "isometry" is in this context?

"the correspondence can be extended to a non-relativistic holography by taking background geometries in General Relativity that have non-relativistic isometries like Schrödinger or Lifshitz symmetries, which are the typical symmetries of a wide class of field theories describing non-relativistic phenomena."

My knowledge is this: in group theory, two groups are homomorphic, in topology, two spaces are homeomorphic, and in geometry, two geometries are isometric. My topology instructor said this once, but I have no formal geometry instruction outside of a GR course and so don't fully understand it.



Isometry is a transformation that preserves metric, as is obvious from the word itself.

And I believe that the quote in your question contains a typo. The phrase should be read as

… geometries in General Relativity that have non-relativistic isometries-like Schrödinger or Lifshitz symmetries …

(Note the dash). So the “isometries-like symmetries” are the symmetries that are similar to isometries in their role, but are not actually isometries. That makes much more sense since Schrödinger [1] and Lifshitz [2] symmetries are symmetries of a field theory but not of the metric (since they include time and space coordinate scaling transformation).

  1. D. Son, “Toward an AdS/cold atoms correspondence: A Geometric realization of the Schrodinger symmetry,” doi, arXiv:0804.3972.

  2. S. Kachru, X. Liu, and M. Mulligan, “Gravity Duals of Lifshitz-like Fixed Points,” doi, arXiv:0808.1725.


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