Abstract
Holographic Conformal Field Theories (CFTs) are usually studied in a limit where the gravity description is weakly coupled. By contrast, lattice quantum field theory can be used as a tool for doing computations in a wider class of holographic CFTs where gravity remains weak but nongravitational interactions {\it in AdS} become strong. We take preliminary steps for studying such theories on the lattice by constructing the discretized theory of a scalar field in AdS$_2$ and investigating its approach to the continuum limit in the free and perturbative regimes. Our main focus is on finite sub-lattices of maximally symmetric tilings of hyperbolic space. Up to boundary effects, these tilings preserve the triangle group as a large discrete subgroup of AdS$_2$, but have a minimum lattice spacing that is comparable to the radius of curvature of the underlying spacetime. We quantify the effects of the lattice spacing as well as the boundary effects, and find that they can be accurately modeled by modifications within the framework of the continuum limit description. We also show how to do refinements of the lattice that shrink the lattice spacing at the cost of breaking the triangle group symmetry of the maximally symmetric tilings.
Highlights
Our expectations about what kinds of behavior are possible in physical systems are often strongly affected by the tractable examples we have at our disposal
One of the strengths of AdS=Conformal field theories (CFTs) is that the conformal symmetry of the field theory dual is built into the spacetime isometries of the AdS description for any values of the bulk parameters, so the boundary theory is automatically conformal
If one is content to work with effective theories in AdS, one can thereby scan over large classes of CFTs with many continuously tuneable
Summary
Our expectations about what kinds of behavior are possible in physical systems are often strongly affected by the tractable examples we have at our disposal. These classes can be a fantastic source of concrete models for many types of strongly coupled physics, and in many cases have eventually led to a deeper understanding of the behavior in the field theory which can after the fact be formulated without reference to a bulk dual. Most known physical systems have critical exponents that are Oð1Þ, which translates to their AdS duals having fields with masses m comparable to the AdS radius Their Compton wavelength is generally spread out over sizes similar to l and perhaps even the tilings without refinement may give good approximations to boundary CFT observables.
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