nLab Dragon's theorem







Dragon’s theorem [Dragon 1979] says that in super Cartan geometry of bosonic dimension 4\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 3\leq 3, where the curvature tensor may contain components not governed by the torsion: “Dragon windows” [Cederwall, Gran & Nilsson 2011]


The original theorem:

implicitly stated there in bosonic dimension = 4.

The evident generalization to higher dimensions is made explicit in:

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

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

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

Last revised on March 19, 2024 at 15:44:31. See the history of this page for a list of all contributions to it.