nLab L-brane

Idea

By reference to the superembedding formalism for super p-branes, the existence of “$L_p$-branes” has been argued [Howe & Sezgin 1997, p. 8], for small natural numbers $p$, with the following properties:

The $p+4$-dimensional target spacetime $X^{p+4}$ carries flux densities

$$\begin{aligned} G_2 & \in \Omega^2_{dR}\big(X^{p+4}\big) \\ G_{p+1} & \in \Omega^{p+1}_{dR}\big(X^{p+4}\big) \\ G_{p+2} & \in \Omega^{p+2}_{dR}\big(X^{p+4}\big) \mathrlap{\,,} \end{aligned}$$

subject to the Bianchi identities

$$\begin{aligned} \mathrm{d} G_2 & = 0 \\ \mathrm{d} G_{p+1} & = 0 \\ \mathrm{d} G_{p+2} & = G_2 \wedge G_{p+1} \mathrlap{\,;} \end{aligned}$$

and on the brane’s worldvolume

$$\Sigma^{p+1} \xrightarrow{\phantom{-}\phi\phantom{-}} X^{p+4}$$

there is a flux density

$$\begin{aligned} F_p & \in \Omega^p_{dR}(X^{p+1}) \end{aligned}$$

satisfying the Bianchi identity

$$\begin{aligned} \mathrm{d} \, F_p & = \phi^\ast G_{p+1} \,. \end{aligned}$$

In Howe, Raetzel & Sezgin 1998, p 3 this is claimed for $p \in \{1,2,3,4,5\}$. The followup article Howe, Raetzel & Rudychev 1999 considers it for $p \in \{3,4,5\}$.

Examples

$L_1$-brane

For $p = 1$ the above Bianchi identities read:

$$\begin{aligned} \mathrm{d} G_2 & = 0 \\ \mathrm{d} G_3 & = G_2 \wedge G_2 \\ \mathrm{d} H_1 & = \phi^\ast G_2 \mathrlap{\,.} \end{aligned}$$

The first two are the Bianchi identities of 5D Maxwell-Chern-Simons theory, hence of the gauge sector of minimal D=5 supergravity.

There is a close structural analogy between this D=5 supergravity on one hand and D=11 supergravity on the other. The latter has Bianchi identities (for the supergravity C-field flux densities $G_4$ and $G_7$) of this form:

$$\begin{aligned} \mathrm{d} G_4 & = 0 \\ \mathrm{d} G_7 & = G_4 \wedge G_4 \\ \mathrm{d} H_3 & = \phi^\ast G_4 \mathrlap{\,,} \end{aligned}$$

where in the last line $H_3$ is the chiral flux on the M5-brane worldvolume.

This makes the $L_1$-brane structure above look like it matches the probe brane version of the known black string of D=5 supergravity (for which cf. Mizoguchi & Ohta 1998, Fujii, Kemmoku & Mizoguchi 2000).

References

Original articles:

• Paul S. Howe, Ergin Sezgin, p. 8 of: Superbranes, Phys. Lett. B 390 (1997) 133-142 [arXiv:hep-th/9607227, doi:10.1016/S0370-2693(96)01416-5]

• Paul S. Howe, O. Raetzel, Ergin Sezgin, pp. 3-4 of: On Brane Actions and Superembeddings, JHEP 9808 (1998) 011 [arXiv:hep-th/9804051, doi:10.1088/1126-6708/1998/08/011]

• Paul S. Howe, O. Raetzel, I. Rudychev, Ergin Sezgin: L-branes, Class. Quant. Grav. 16 (1999) 705-722 [arXiv:hep-th/9810081, doi:10.1088/0264-9381/16/3/006]

Review:

• Steven Duplij: L-Brane, in: Concise Encyclopedia of Supersymmetry, Springer (2017) 225 [doi:10.1007/1-4020-4522-0_294, pdf]