Abstract
The Holographic Wess-Zumino (HWZ) consistency condition is shown through a step by step mapping of renormalization group flows to Hamiltonian systems, to lead to the Holographic anomaly. This condition codifies how the energy scale, when treated as the emergent bulk direction in Holographic theories, is put on equal footing as the other directions of the space the field theory inhabits. So, this is a defining feature of theories possessing local Holographic bulk duals. In four dimensional Holographic conformal field theories, the a and c anomaly coefficients are equated, and this is seen as a defining property of theories which possess General Relativity coupled to matter as a dual. Hence, showing how the former consistency condition leads to the latter relation between anomaly coefficients adds evidence to the claim that the HWZ condition is a defining feature of theories possessing local gravity duals.
Highlights
When the bulk theory is expected to be just classical general relativity coupled to matter fields, the relevant subset of these holographic theories are in a regime characterised by possessing a large number of degrees of freedom in which all the operators barring a finite set gain an infinitely large anomalous dimension
The Holographic Wess-Zumino (HWZ) consistency condition is shown through a step by step mapping of renormalization group flows to Hamiltonian systems, to lead to the Holographic anomaly
Restricting attention to the case where the dynamics of the bulk theory is dictated by general relativity, to be borne out, it is necessary to ensure that diffeomorphism invariance in the bulk emerges
Summary
The following work is an amalgamation of several related ideas and in order to point out what the novelty of the results are, it will help to put it in context with earlier work. The novelty of the results presented in this article is to start with deducing the off shell bulk theory of general relativity coupled to matter by combine the insights of mapping RG flows of large N theories to Hamiltonian systems and that the emergence of general covariance as encoded in the so called Holographic Wess-Zumino consistency conditions is a defining feature of holographic dualities. This approach sidesteps the path integral construction in the quantum RG as it applies mainly to the large N limit. The Hamiltonian approach to holographic RG can be applied in order to study the theory at UV fixed point leading to identical results of conventional Holographic renormalization in the AdS/CFT context
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