$c$        $a$            $c-a$
  ------------------------------------ ----- --------------- ----------------
   Massless $\mathcal{N}=8$ multiplet   $0$   $\frac{5}{2}$   $-\frac{5}{2}$

  : Central charges $c$ and $a$ for ${\cal N}=8$ supergravity.[]{data-label="table:centralcharges8"}

Interpretation of Quantum Corrections
-------------------------------------

The anomalous contribution to the trace of the energy momentum tensor is independent of position because spacetime is homogeneous. A classical cosmological constant in the action similarly gives a constant contribution but the origin of the anomalous contribution is a divergence $W_{\rm div} = {1\over 2} D_0\ln \epsilon^2/\ell^2_0$ in the effective action that manifests itself in the renormalized action as a term that evolves logarithmically $$W_{\rm ren} = -  {1\over 2}D_0\ln {x^2_{\rm phys}\over\ell^2_0}~.$$ The renormalization scale $\ell_0$ enters as an IR cutoff on the integral over the heat kernel. It is arbitrary but of order of the AdS-scale. The physical length scale $x_{\rm phys}$ depends on the process as usual and may be anywhere in the range from much smaller than the AdS scale (for UV processes) to much larger than the AdS scale (for the IR properties).

We interpret the scale dependent quantum effective action as a contribution $\delta\Lambda$ to the cosmological constant determined by $$W_{\rm ren}  = - {{\textrm{Vol}_{\textrm{AdS}_4}}\delta\Lambda \over 8\pi G}~.$$ It is convenient to express the running in 