nLab D=4 supergravity Lie 2-algebra

Idea

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 $N = 2$ 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 $\propto \overline{\psi} \wedge \Gamma^a \psi \wedge e_a$ (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 $\propto \overline{\psi} \wedge \Gamma^{a b} \psi \wedge e_a \wedge e_b$. 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))

Related concepts

• supergravity Lie 3-algebra

• supergravity Lie 6-algebra

References

Background discussion (and further pointers) for $p$-form fields in D=4 supergravity is in the introduction of

• Murat Gunaydin, S. McReynolds, M. Zagermann, Unified $N=2$ Maxwell-Einstein and Yang-Mills-Einstein Supergravity Theories in Four Dimensions, JHEP 0509:026,2005 (arXiv:hep-th/0507227)

The D=4 supergravity Lie 2-algebra was given in

• Laura Andrianopoli, Riccardo D'Auria, Luca Sommovigo, $D=4$, $N=2$ Supergravity in the Presence of Vector-Tensor Multiplets and the Role of higher p-forms in the Framework of Free Differential Algebras, Adv.Stud.Theor.Phys.1:561-596,2008 (arXiv:0710.3107)

Further discussion is in

• Laura Andrianopoli, Riccardo D'Auria, Luca Sommovigo, Mario Trigiante, $D=4$, $N=2$ Gauged Supergravity coupled to Vector-Tensor Multiplets, Nucl.Phys.B851:1-29,2011 (arXiv:1103.4813)