Background independent holographic description : From matrix field theory to quantum gravity

TL;DR

The work addresses the challenge of deriving holographic duals for general quantum field theories in a background-independent way. It introduces a local renormalization group with a spacetime-dependent scale and promotes high-energy sources to dynamical fields, yielding a $(D+1)$-dimensional quantum gravity with matter as the bulk dual of a $D$-dimensional matrix field theory. A key insight is that the bulk is governed by $(D+1)$ first-class constraints that generate bulk diffeomorphisms, with the RG flow realizing a Hamiltonian evolution in the emergent dimension. The approach provides a first-principles route to holographic duals, clarifies the role of multi-trace operators via bulk dynamics, and offers a controlled setting to explore gravity emergence in large-$N$ theories, including simple explicit examples and a clear contrast with conventional RG. Overall, it lays out a concrete framework to study background-independent holography and quantum RG in strongly coupled systems.

Abstract

We propose a local renormalization group procedure where length scale is changed in spacetime dependent way. Combining this scheme with an earlier observation that high energy modes in renormalization group play the role of dynamical sources for low energy modes at each scale, we provide a prescription to derive background independent holographic duals for field theories. From a first principle construction, it is shown that the holographic theory dual to a D-dimensional matrix field theory is a (D+1)-dimensional quantum theory of gravity coupled with matter fields of various spins. The gravitational theory has (D+1) first-class constraints which generate local spacetime transformations in the bulk. The (D+1)-dimensional diffeomorphism invariance is a consequence of the freedom to choose different local RG schemes.

Figures (4)

• Figure 1: Two ways of generating quantum corrections to the linear order in $dz$. Each circle denotes trace of a chain of matrix fields. Solid lines represent chains of low energy fields and each dashed line represents a high energy field. (a) Contraction of a pair of high energy fields within a single-trace operator generates two singe-trace operators (the first and the third) and one double-trace operator (the second). In the large $N$ limit, only the second term is $O(N^2)$. (b) At the quadratic order, one can fuse two single-trace operators each of which contains one high energy mode. This leads to one double-trace operator and two single-trace operators, all of which are $O(N^2)$.
• Figure 2: The $D$-dimensional coordinate of the low energy mode is shifted infinitesimally relative to the coordinate of the field defined at high energy.
• Figure 3: (a) Bulk spacetime made of the $D$-dimensional boundary spacetime and the semi-infinite line that represents the length scale in the RG procedure. Each step of coarse graining, say the $l$-th step, generates a set of $D$-dimensional fields $\left( J^{(l)n}(x), P^{(l)}_n(x) \right)$ that represent dynamical sources and operators at that scale. These fields are combined into $(D+1)$-dimensional fields $\left( J^{n}(x,z), P_n(x,z) \right)$ in the bulk, where the extra coordinate is given by $z = l dz$. Each 'vertical' line traces the positions of the bulk fields which are generated from the original field variable $\Phi(x)$ at each $x$ in the boundary spacetime. The spacetime dependent shift $N^{\mu}(x,z)$ causes the bulk fields to have different $D$-dimensional coordinates from that of $\Phi(x)$. Each 'horizontal' line represents the manifold in the bulk spacetime with an equal $z$ coordinate. Because the speeds of coarse graining are in general different at different points in spacetime, two points within the manifold with an equal $z$ do not in general have the same proper length along the extra dimension, where the proper length is the scale in the RG. (b) The same bulk spacetime where the coordinate $z$ is used instead of the proper length along the extra dimension. The vertical lines have the same meaning as in (a). Each horizontal line represents the manifold with an equal proper length, that is, the set of points with the same length scale in RG. Note that an horizontal line that is concave upward in (a) is concave downward in (b).
• Figure 4: (a) In the conventional RG, multi-trace operators are generated at low energies although only single-trace operators are turned on at UV. Once the initial condition is given, there is a unique RG trajectory determined by the classical beta function. (b) In the holographic description, one only needs to keep track of single-trace operators under the RG flow at the expense of making the sources for the single-trace operators dynamical variables. The partition function is given by sum over all possible RG trajectories for the single-trace operators. One has the freedom to employ different RG schemes, namely, one can choose different 'speed' of RG flow at different scales. This freedom corresponds to the diffeomorphism invariance in the bulk.