∞-Lie theory (higher geometry)
Background
Smooth structure
Higher groupoids
Lie theory
∞-Lie groupoids
∞-Lie algebroids
Formal Lie groupoids
Cohomology
Homotopy
Related topics
Examples
-Lie groupoids
-Lie groups
-Lie algebroids
-Lie algebras
Formalism
Definition
Spacetime configurations
Properties
Spacetimes
black hole spacetimes | vanishing angular momentum | positive angular momentum |
---|---|---|
vanishing charge | Schwarzschild spacetime | Kerr spacetime |
positive charge | Reissner-Nordstrom spacetime | Kerr-Newman spacetime |
Quantum theory
superalgebra and (synthetic ) supergeometry
D=4 supergravity is often formulated as a theory of gravity coupled to scalar-vector multiplets only, i.e. 1-form gauge fields. On the other hand, when the theory is thought of as obtained by KK-compactification from D=11 supergravity, then it naturally contains also 2-form fields (tensor multiplets), i.e. higher gauge fields. If these are massless, then in 4-dimensions they may be dualized (via Hodge duality of their field strengths) to scalars, which is why they often do not explicitly appear in the formulation. However, when they are massive, which happens for instance when the higher dimensional theory is reduced via a flux compactification, then such dualization does not apply (at least not so directly) and the 2-form higher gauge fields need to be made explicit (see the introduction of Gunyadin-McReynolds-Zagerman 05).
In the D'Auria-Fré formulation of supergravity such higher form field contributions are reflected by L-infinity algebra extensions of the super Minkowski spacetime supersymmetry super Lie algebra (traditionally displayed in terms of dual Chevalley-Eilenberg algebras, called “FDA”s in the supergravity literature).
For the 2-form fields of D=4 supergravity this yields a Lie 2-algebra (Andrianopoli-D’Auria-Sommovigo 07 (4.1)-(4.7)), which hence might be called the “D=4 supergravity Lie 2-algebra”. In fact, including the moduli fields, this is a Lie 2-algebroid (Andrianopoli-D’Auria-Sommovigo 07 (4.8)-(4.9)). See also (AAST 11, (4.1)-(4.9)).
This Lie 2-algebra is a non-abelian variant of the L-infinity extension classified by the 3-cocycle (Andrianopoli-D’Auria-Sommovigo 07 (4.5)), which is the WZW term for the Green-Schwarz superstring in 4d (see the notation and conventions at super-Minkowski spacetime).
But the brane scan says that there is also a super 2-brane in 4d whose WZW-term is the 4-cocycle . Accordingly there should also be a 4d supegravity Lie 3-algebra. This is left to “future investigations” in (Andrianopoli-D’Auria-Sommovigo 07, p. 19), but the relevant extension formula by the 4-cocycle is shown in (Andrianopoli-D’Auria-Sommovigo 07 (2.24))
Background discussion (and further pointers) for -form fields in D=4 supergravity is in the introduction of
The D=4 supergravity Lie 2-algebra was given in
Further discussion is in
Last revised on July 17, 2024 at 19:58:34. See the history of this page for a list of all contributions to it.