The Holographic Geometry of the Renormalization Group and Higher Spin Symmetries

Abstract

We consider the Wilson-Polchinski exact renormalization group applied to the generating functional of single-trace operators at a free-fixed point in $d=2+1$ dimensions. By exploiting the rich symmetry structure of free field theory, we study the geometric nature of the RG equations and the associated Ward identities. The geometry, as expected, is holographic, with $AdS$ spacetime emerging correspondent with RG fixed points. The field theory construction gives us a particular vector bundle over the $d+1$-dimensional RG mapping space, called a jet bundle, whose structure group arises from the linear orthogonal bi-local transformations of the bare fields in the path integral. The sources for quadratic operators constitute a connection on this bundle and a section of its endomorphism bundle. Recasting the geometry in terms of the corresponding principal bundle, we arrive at a structure remarkably similar to the Vasiliev theory, where the horizontal part of the connection on the principal bundle is Vasiliev's higher spin connection, while the vertical part (the Faddeev-Popov ghost) corresponds to the $S$-field. The Vasiliev equations are then, respectively, the RG equations and the BRST equations, with the RG beta functions encoding bulk interactions. Finally, we remark that a large class of interacting field theories can be studied through integral transforms of our results, and it is natural to organize this in terms of a large $N$ expansion.

44 pages, 4 figures; v2 contains additional references, elimination of typos and tightening up of constructions

References in corpus (6)

• AdS_4/CFT_3 Construction from Collective Fields
• Effective action in a higher-spin background
• Holographic dual of free field theory
• Holographic description of quantum field theory
• Holographic description of large N gauge theory
• An action for higher spin gauge theory in four dimensions

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