nLab Dragon's theorem

Idea

Dragon’s theorem [Dragon 1979] says that in super Cartan geometry of bosonic dimension $\geq 4$ with structure group the Lorentzian spin group:

1. the curvature-tensor is a linear function of the torsion-tensor (i.e., of the full super torsion tensor),

2. expressed this way, the curvature-Bianchi identity is implied already by the torsion Bianchi identity.

This observation arose in and finds application in supergravity, whose fields are Cartan super frame fields and whose equations of motion are typically controlled by rheonomy/torsion constraints, so that finding on-shell field histories is equivalent to solving the torsion- & curvature Bianchi identities subject to these constraints. Here the curvature Bianchi identity typically looks much more involved than the torsion Bianchi identity, but Dragon’s theorem clarifies that it need not actually be solved independently.

Dragon’s theorem fails in bosonic dimension $\leq 3$, where the curvature tensor may contain components not governed by the torsion: “Dragon windows” [Cederwall, Gran & Nilsson 2011]

References

The original theorem:

• Norbert Dragon, Torsion and curvature in extended supergravity, Zeitschrift für Physik C – Particles and Fields 2 (1979) 29–32 [doi:10.1007/BF01546233, pdf]

implicitly stated there in bosonic dimension = 4.

The evident generalization to higher dimensions is made explicit in:

• Alexander W. Smith, Torsion and curvature in higher dimensional supergravity theories, Zeitschrift für Physik C – Particles and Fields 24 (1984) 85–86 [doi:10.1007/BF01576291]

Proof of the first part of the statement is also in:

• John Lott, Prop. 7 in: Torsion constraints in supergeometry, Comm. Math. Phys. 133 (1990) 563-615 [doi:10.1007/BF02097010]

Articles invoking Dragon’s theorem in higher-dimensional supergravity (D=10 and D=11):

• Eugene Cremmer, Sergio Ferrara, p. 63 in: Formulation of Eleven-Dimensional Supergravity in Superspace, Phys. Lett. B 91 (1980) 61 [doi:10.1016/0370-2693(80)90662-0]

• Loriano Bonora, M. Bregola, Kurt Lechner, Paolo Pasti, Mario Tonin, p. 884 in: Anomaly-free supergravity and super-Yang-Mills theories in ten dimensions, Nuclear Physics B 296 4 (1988) [doi:10.1016/0550-3213(88)90402-6, inspire:247764]

• A. Candiello, Kurt Lechner, Duality in Supergravity Theories, Nucl. Phys. B 412 (1994) 479-501 [arXiv:hep-th/9309143, doi:10.1016/0550-3213(94)90389-1]

Discussion of the effects in low dimensions where the theorem does not hold:

• Martin Cederwall, Ulf Gran, Bengt E. W. Nilsson, $D=3$, $N=8$ conformal supergravity and the Dragon window, JHEP 1109:101 (2011) [arXiv:1103.4530, doi:10.1007/JHEP09(2011)101]