Triangle anomalies from Einstein manifolds

Abstract

The triangle anomalies in conformal field theory, which can be used to determine the central charge $a$, correspond to the Chern-Simons (CS) couplings of gauge fields in $\mathrm{AdS}_5$ under the gauge$/$gravity correspondence. We present a simple geometrical formula for the CS couplings in the case of type IIB supergravity compactified on a five-dimensional Einstein manifold $X$. When $X$ is a circle bundle over del Pezzo surfaces or a toric Sasaki-Einstein (SE) manifold, we show that the gravity result is in perfect agreement with the corresponding quiver gauge theory. Our analysis reveals an interesting connection with the condensation of giant gravitons or dibaryon operators which effectively induces a rolling among SE vacua.