In this paper, we revisit scalar field theories in $d$ space-time dimensions possessing $U(N)$ global symmetry. Following our recent work [1], we consider the generating function of correlation functions of all $U(N)$-invariant, single-trace operators at the free-fixed point. The exact renormalization group equations are cast as Hamilton equations of radial evolution in a model space-time of one higher dimension, in this case ${\mathrm{AdS}}_{d+1}$. The geometry associated with the renormalization group equations is seen to emerge naturally out of the infinite jet bundle corresponding to the field theory and suggests their interpretation as higher-spin equations of motion. While the higher-spin equations we obtain are remarkably simple, they are nonlocal in an essential way. Nevertheless, solving these bulk equations of motion in terms of a boundary source, we derive the on-shell action and demonstrate that it correctly encodes all of the correlation functions of the field theory, written as “Witten diagrams.” Since the model space-time has the isometries of the fixed point, it is possible to construct new higher-spin theories defined in terms of geometric structures over other model space-times. We illustrate this by explicitly constructing the higher-spin renormalization group equations corresponding to the $z=2$ nonrelativistic free field theory in $D$ spatial dimensions. In this case, the model space-time is the Schrödinger space-time, ${\text{Schr}}_{D+3}$.

• Received 3 November 2014

DOI:https://doi.org/10.1103/PhysRevD.91.026002