Weyl Fermion         $\frac{1}{40}$      $\frac{11}{720}$     $\frac{7}{720}$
          Vector            $\frac{1}{10}$      $\frac{31}{180}$    -$\frac{13}{180}$
   Antisymmetric Tensor    $\frac{1}{120}$     $-\frac{179}{360}$    $\frac{91}{180}$
        Gravitino         $-\frac{411}{360}$   $-\frac{589}{720}$   $-\frac{233}{720}$
         Graviton         $\frac{783}{180}$    $\frac{571}{180}$     $\frac{53}{45}$

  : Central charges $c$ and $a$ for minimally coupled massless fields. Each entry is a physical field with two degrees of freedom except the scalar, which has just one degree of freedom.[]{data-label="table:centralcharges"}

Computations in AdS$_4$
-----------------------

We now revisit these computations in the context of AdS$_4$. This geometry is conformally flat so the Weyl tensor vanishes and therefore the central charge $c$ plays no role. Our focus on $a$ is complementary to techniques that impose Einstein’s equations in vacuum and identify just the Riemann-squared terms which have coefficient $c-a$.

The natural representations for fields in AdS$_4$ are the symmetric, transverse, and traceless (STT) tensors with spin $s$. The heat kernels for these fields were comprehensively analyzed by Camporesi and Higuchi [@Camporesi:1992wn; @Camporesi:1994ga; @Higuchi:1986wu] (and recently developed further [@Giombi:2014yra]) both using explicit mode functions and also using group theory. We present their results for the AdS$_4$ heat kernel of a massive spin $s$ field with conformal dimension $\Delta$ as $$\label{eq:KAmaster}
K^{(s,\nu)}(t) = \frac{1}{\ell_A^4}\int_0^\infty d\lambda ~\mu_s(\lambda) ~e^{-\frac{t}{\ell_A^2} (\lambda^2+\nu^2)}~,$$ where $\nu^2 = (\Delta - \frac{3}{2})^2$. The conformal dimension $\Delta$ is equivalent 