Casimir energy on the sphere and 6D CFT trace anomaly
We elucidate the dependence of the Casimir energy on the trace anomaly coefficients for a six-dimensional CFT on $R\times S^5$. This extends the universal dependence on the central charge in 2D and the relation by Cappelli and Coste in 4D, unveiling the role of the trivial total derivatives in the anomaly that renders the Casimir energy scheme dependent. We obtain $$E_o=-\frac{15}{8}\,a_6 -\frac{5}{12}\,\left(g_5+\frac{1}{4}\,g_7+\frac{1}{2}\,g_8-10\, g_9+g_{10}\right),$$ with $a_6$ being the type A central charge and the $g$'s, the coefficients of five out of six terms that form a basis for trivial total derivatives. The derivation is based on the Polyakov formulas (conformal primitive) resulting from the integration of the trace anomaly. Alternatively, on a 6D conformally flat background the above basis is redundant and one can simplify further to get, in terms of the Schouten scalar $J$ and the Schouten tensor $V$, Branson's basis for trivial total derivatives $\nabla^2\nabla^2 J$, $\nabla^2J^2$ and $\nabla^2|V|^2+2\nabla\cdot(V\cdot\nabla\,J)$ with coefficients $γ_1, γ_2$ and $γ_3$, respectively, \begin{equation} \nonumber E_o=-\frac{15}{8}a_6-\frac{1}{24}\left(γ_1-γ_2 -\frac{1}{8}γ_3\right)~. \end{equation}