Generalised Calibrations in AdS backgrounds from Exceptional Sasaki-Einstein Structures

We study generalised calibrations described by Exceptional Sasaki-Einstein structures on AdS backgrounds in type IIB and M-theory, placing the focus on the generalised Reeb vectors. The inequalities for the energy bound are derived by decomposing a $\kappa$-symmetry condition and equivalently, bispinors in calibration conditions from existing literature. We explain how the closure of the calibration forms is related to the integrability conditions of the Exceptional Sasaki-Einstein structure, in particular for AdS space-filling or point-like branes. Doing so, we show that the form parts of the twisted vector structure in M-theory provides generalised calibrations, reproducing the result conjectured in a recent paper. The IIB case yields analogous results. Finally, we discuss briefly the extension of the conjecture to hypermultiplet structures and other brane configurations.