Non-Relativistic Holography -- A Group-Theoretical Perspective

Abstract: We give a review of some group-theoretical results related to non-relativistic holography. Our main playgrounds are the Schr\"odinger equation and the Schr\"odinger algebra. We first recall the interpretation of non-relativistic holography as equivalence between representations of the Schr\"odinger algebra describing bulk fields and boundary fields. One important result is the explicit construction of the boundary-to-bulk operators in the framework of representation theory, and that these operators and the bulk-to-boundary operators are intertwining operators. Further, we recall the fact that there is a hierarchy of equations on the boundary, invariant w.r.t. Schr\"odinger algebra. We also review the explicit construction of an analogous hierarchy of invariant equations in the bulk, and that the two hierarchies are equivalent via the bulk-to-boundary intertwining operators. The derivation of these hierarchies uses a mechanism introduced first for semi-simple Lie groups and adapted to the non-semisimple Schr\"odinger algebra. These requires development of the representation theory of the Schr\"odinger algebra which is reviewed in some detail. We also recall the $q$-deformation of the Schr\"odinger algebra. Finally, the realization of the Schr\"odinger algebra via difference operators is reviewed.