Musings

What I didn’t know was that this bound was related to familiar bounds on the central charges $a,c$ of the SCFT. In a curved background, these are given by $$\tensor{T}{_{\mu}_^{\mu}} = \frac{c}{16\pi^2} {(\text{Weyl})}^2 -\frac{a}{16\pi^2} (Euler)$$ where $$\begin{aligned} {(\text{Weyl})}^2 &= R_{\mu\nu\lambda\rho}^2 -2 R_{\mu\nu}^2 +\tfrac{1}{3} R^2\\ (\text{Euler}) &= R_{\mu\nu\lambda\rho}^2 -4 R_{\mu\nu}^2 + R^2 \end{aligned}$$ In a supersymmetric gauge theory, the coefficients, $a$, $c$ are one-loop exact, being related by supersymmetric Ward identities to certain R-current anomalies. For an $\mathcal{N}=2$ gauge theory, with gauge group, $\mathcal{G}$, and hypermultiplets in representation $R_{\text{hyper}}$,

and, for an $\mathcal{N}=1$ gauge theory, with gauge group, $\mathcal{G}$, and chiral multiplets in representation $R_{\text{chiral}}$,

In each case, $dim(R)$ and $dim(adj)$ are non-negative numbers. Hence we have inequalities

for $\mathcal{N}=2$ and

for $\mathcal{N}=1$.

While I stated the results for supersymmetric gauge theories, the bounds (3) and (4) follow from superconformal Ward identities, as shown in Appendix C of Hofman and Maldacena (see also Shapere and Tachikawa). For $\mathcal{N}=1,2$, these bound are saturated by free field theories (the lower bound, by free abelian vector multiplets, the upper bound by free massless chiral/hyper multiplets). If this continues to hold for $\mathcal{N}=0$, we can write

where the lower limit comes from free scalar field theory and the upper limit from free Maxwell theory.

The connection that I didn’t make was that $a,c$ also depend on the coefficient of the curvature-squared terms in the bulk theory and, for Gauss-Bonnet gravity, one has

so that the lower limit $a/c\geq 1/2$ corresponds to $\lambda_{\text{GB}}\leq 9/100$, as above.