Gaugino condensation plays a crucial role in the formation of de Sitter vacua. However, the theory of this effect is still incomplete. In four dimensions, the perfect square nature of gaugino couplings follows from the square of auxiliaries in the supergravity action. We explain here why the supersymmetric non-Abelian Dp-brane action, which is the basis for the theory of ten-dimensional gaugino condensation, must have a four-gaugino coupling. This term in the Einstein-Yang-Mills ten-dimensional supergravity is a part of the perfect square, mixing a 3-form with a gaugino bilinear. The perfect square term follows from the superspace geometry, being a square of the superspace torsion. The supercovariant equation of motion for a gaugino on the non-Abelian Dp-brane also involves a supertorsion in agreement with the perfect square term in the action.

• Received 30 January 2019

DOI:https://doi.org/10.1103/PhysRevD.99.066003

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Published by the American Physical Society