The AdS/CFT dictionary and its parameter regimes

Take the interacting boundary gauge theory to have gauge group SU(N) and sixteen Poincaré supercharges, conventionally called four-dimensional N equals four supersymmetry. The decoupled center-of-mass U(1) from the brane construction is treated separately. The full bulk has ten dimensions: five in the AdS factor and five compact sphere directions. Reducing on the sphere gives an effective five-dimensional gravity description with a tower of additional modes, rather than making the boundary five dimensional.

Use natural units with reduced Planck constant, light speed and Boltzmann constant set to one. Define the Yang-Mills coupling convention explicitly so that its square equals four pi times the string coupling. Sources use different trace conventions, and translating that factor is necessary before comparing a numerical ’t Hooft coupling. The radius-to-string-length relation then gives a dimensionless curvature parameter. Matching symmetries and parameter relations supports the proposed correspondence but is not a proof of every dynamical statement.

In d plus one effective bulk dimensions, use the Einstein action with coefficient inverse sixteen pi times Newton constant. Newton constant has length to the d minus one units in these natural conventions, making the action dimensionless. Boundary terms are needed for a well-defined variational problem and renormalization. Omitting those terms from a schematic bulk expression does not authorize dropping them when computing a free energy or a boundary correlation function.

For the AdS5 example, dimensional reduction and flux quantization give a leading Newton coefficient proportional to inverse N squared in units of the AdS radius. This controls the gravitational loop expansion at fixed low energies. Finite string length produces a distinct higher-derivative expansion, whose actual leading power depends on the theory and observable. An estimate of one small ratio is therefore not a universal error bar for every correlator or a guarantee that an arbitrary truncated bulk action is accurate.

A sufficient simple supergravity regime has large N, large ’t Hooft coupling and small string coupling. For example, keeping the coupling much larger than one while much smaller than N can meet both curvature and string-loop conditions up to the stated numerical factors. The boundary theory is then strongly coupled, so ordinary weak-coupling Feynman expansions are generally not simultaneous approximations to the same observable. Protected quantities can be compared across regimes, but that protection must be established.

A top-down model derives fields and interactions from a specified string construction. A bottom-up model instead chooses a gravitational action designed to capture selected field-theory features; it must justify stability, boundary conditions and its approximation range independently. Neither category automatically describes an actual laboratory material or cosmology. The quadratic lab can illustrate an inverse-color-number-squared scaling after a declared variable mapping, but it has no string coupling, compactification or quantum gravitational loop computation.