Casimir energy on the sphere and 6D CFT trace anomaly

TL;DR

This paper explores how the Casimir energy in six-dimensional conformal field theories depends on trace anomaly coefficients, extending known relations from lower dimensions and clarifying the scheme dependence due to trivial total derivatives.

Contribution

It provides a detailed derivation of the Casimir energy's dependence on trace anomaly coefficients in 6D CFTs, including universal formulas and simplifications on conformally flat backgrounds.

Findings

Derived explicit formula for Casimir energy in terms of anomaly coefficients.

Showed scheme dependence of Casimir energy due to trivial total derivatives.

Simplified the basis of trivial total derivatives on conformally flat backgrounds.

Abstract

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 $\gamma_1, \gamma_2$ and $\gamma_3$, respectively, \begin{equation} \nonumber E_o=-\frac{15}{8}a_6-\frac{1}{24}\left(\gamma_1-\gamma_2 -\frac{1}{8}\gamma_3\right)~. \end{equation}