Non-relativistic metrics from back-reacting fermions

Abstract

It has recently been pointed out that under certain circumstances the back-reaction of charged, massive Dirac fermions causes important modifications to AdS2 spacetimes arising as the near-horizon geometry of extremal black holes. In a WKB approximation, the modified geometry becomes a non-relativistic Lifshitz spacetime. In three dimensions, it is known that integrating out charged, massive fermions gives rise to gravitational and Maxwell Chern–Simons terms. We show that Schrödinger (warped AdS3) spacetimes exist as solutions to a gravitational and Maxwell Chern–Simons theory with a cosmological constant. Motivated by this, we look for warped AdS3 or Schrödinger metrics as exact solutions to a fully back-reacted theory containing Dirac fermions in three and four dimensions. We work out the dynamical exponent in terms of the fermion mass and generalize this result to arbitrary dimensions.