Holographic interpretations of the renormalization group

Abstract

In semiclassical holographic duality, the running couplings of a field theory are conventionally identified with the classical solutions of field equations in the dual gravitational theory. However, this identification is unclear when the bulk fields fluctuate. Recent work has used a Wilsonian framework to propose an alternative identification of the running couplings in terms of non-fluctuating data; in the classical limit, these new couplings do not satisfy the bulk equations of motion. We study renormalization scheme dependence in the latter formalism, and show that a scheme exists in which couplings to single trace operators realize particular solutions to the bulk equations of motion, in the semiclassical limit. This occurs for operators with dimension ∆ / ∈ d 2 + Z, for sufficiently low momenta. We then clarify the relation between the saddle point approximation to the Wilsonian effective action (S W ) and boundary conditions at a cutoff surface in AdS space. In particular, we interpret non-local multi-trace operators in S W as arising in Lorentzian AdS space from the temporary passage of excitations through the UV region that has been integrated out. Coarse-graining these operators makes the action effectively local.

Holographic renormalisation group flows and renormalisation from a Wilsonian perspective

“…Our interpretation of this relation looks quite transparent and explicit to us, but we should point out that it is closely related to previous proposals in [20] and [17], respectively. The first proposal [20] is a perturbative version of our renormalisation scheme, as can be readily checked by a field expansion of our equations.…”

“…Nevertheless, we should point out that below the unitarity bound singularities may arise at certain values of the momenta [17]. With this W , (4.6) has a unique solution, at least in some neighbourhood of the fixed point.…”

“…Observe that S B t and W are, respectively, the result of the UV and IR integrations at the classical level, and this extremization corresponds to the remaining integration over ϕ [17,18,20]. Note as well that (4.6) can be understood as the requirement of compatibility of the boundary conditions imposed on the classical fields by S B t and W .…”

“…As nicely explained in ref. [17], this is not yet a truly Wilsonian approach, as this Wilson action depends on physics below the IR cutoff. In ref.…”

“…This can be achieved by taking as h b t 0 an arbitrary solution to the equations of motion (including dilatations and with t −1 0 the dimensionless radial coordinate), for any non-singular IR condition [22]. 17 The reason is that the UV Dirichlet condition will then give rise to the same solution for any t 0 . The limit will then be trivially finite.…”