kernels (\[eq:Kbosemassless\]) and (\[eq:Kfermimassless\]) correspond to power law divergences in the effective action. The boson and fermion contributions to these divergenes cancel when $$\label{eq:ssumrule}
\sum (-)^{2s} s^n = 0~,$$ for $n=0,2$. The massless spectrum only comprises maximal helicity where $|h|=s$ so this condition is equivalent to the helicity sum rule (\[eq:supertrace\]) for $n=0,2$. This is satisfied for ${\cal N}\geq 3$ supergravity and we will focus on these theories.

     **Spin**      **Conformal Dimension** $\Delta$   $SO(8)$ **Multiplicity**
  --------------- ---------------------------------- --------------------------
        $2$                      $3$                            $1$
   $\frac{3}{2}$            $\frac{5}{2}$                       $8$
        $1$                      $2$                            $28$
   $\frac{1}{2}$            $\frac{3}{2}$                       $56$
       $ 0$                      $1$                            $35$
        $0$                      $2$                            $35$

  : The conformal dimensions and multiplicities of the massless multiplet in $\mathcal{N}=8$ supergravity.[]{data-label="table:conformaldimensions"}

For maximal $\mathcal{N}=8$ SUGRA the standard spectrum given in table \[table:conformaldimensions\] satisfies the sum rule (\[eq:ssumrule\]) even for $n=4, 6$ yet the sum of the boson and fermion heat kernels do not vanish $$\label{eq:kn8}
K_{{\cal N}=8}^{\textrm{total}} =  \langle T_\mu^{\mu}\rangle_{\textrm{ren}} = \frac{1}{16\pi^2\ell_A^4}(-60)~.$$ This is possible because the bosonic and fermionic heat kernels (\[eq:Kbosemassless\]-\[eq:Kfermimassless\]) are given different polynomials in the spin $s$.

We can represent the heat kernel result (\[eq:kn8\]) for ${\cal N}=8$ supergravity as an $a$ anomaly for the entire multiplet, $$\label{eq:N=8div}
a_{{\cal N}=8} = \frac{5}{2}~.$$ Considering also the values of $c$ 