NONRELATIVISTIC HOLOGRAPHY - A GROUP-THEORETICAL PERSPECTIVE

We give a review of some group-theoretical results related to nonrelativistic holography. Our main playgrounds are the Schrodinger equation and the Schrodinger algebra. We first recall the interpretation of nonrelativistic holography as equivalence between representations of the Schrodinger 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 with respect to Schrodinger 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 semisimple Lie groups and adapted to the nonsemisimple Schrodinger algebra. These require development of the representation theory of the Schrodinger algebra which is reviewed in some detail. We also recall the q-deformation of the Schrodinger algebra. Finally, the realization of the Schrodinger algebra via difference operators is reviewed.