Equivariant localization for wrapped M5-branes and D4-branes

Following the work of [1], we consider six- and seven-dimensional gauged supergravity coupled to a vector multiplet on an ansatz of $Ad{S}_{4,5}\times M_2$, respectively, where $M_2$ is a two-dimensional surface with a Killing vector. We construct equivariantly closed forms from spinor bilinears of Killing spinors. From the integration of equivariantly closed forms via the Berline-Vergne-Atiyah-Bott fixed point formula, we derive the gravitational blocks which were previously conjectured. For the construction of equivariantly closed forms, we formulate six- and seven-dimensional gauged supergravity in the very special geometry.