# Gaugino Condensation and Geometry of the Perfect Square

## TL;DR

This paper explains the geometric origin of the perfect square structure in gaugino couplings within supergravity, linking it to superspace torsion and providing insights into gaugino condensation in string theory.

## Contribution

It demonstrates that the perfect square nature of gaugino couplings arises from superspace geometry and supertorsion, clarifying the structure of 10d supergravity actions.

## Key findings

- Gaugino couplings form a perfect square in supergravity.
- The perfect square term originates from superspace torsion.
- Supercovariant gaugino equations involve supertorsion.

## Abstract

Gaugino condensation plays a crucial role in the formation of dS vacua. However, the theory of this effect is still incomplete. In 4d 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 10d gaugino condensation, must have a 4-gaugino coupling. This term in the Einstein-Yang-Mills 10d supergravity is a part of the perfect square, mixing a 3-form with gaugino bilinear. The perfect square term follows from the superspace geometry, being a square of the superspace torsion. The supercovariant equation of motion for gaugino on the non-Abelian Dp-brane also involves a supertorsion in agreement with the perfect square term in the action.