On the exceptional generalised Lie derivative for d ≥ 7

Abstract

In this work we revisit the E 8 × R + generalised Lie derivative encoding the algebra of diffeomorphisms and gauge transformations of compactifications of M-theory on eight-dimensional manifolds, by extending certain features of the E 7 × R + one. Compared to its E d × R + , d ≤ 7 counterparts, a new term is needed for consistency. However, we find that no compensating parameters need to be introduced, but rather that the new term can be written in terms of the ordinary generalised gauge parameters by means of a connection. This implies that no further degrees of freedom, beyond those of the field content of the E 8 group, are needed to have a well defined theory. We discuss the implications of the structure of the E 8 × R + generalised transformation on the construction of the d = 8 generalised geometry. Finally, we suggest how to lift the generalised Lie derivative to eleven dimensions.

E8 geometry

“…Hohm and Samtleben [45] nevertheless managed to base a description of 11-dimensional supergravity in a 3+8 split on such transformations, however with the drawback that a geometric understanding was lacking. It was observed by one of the present authors [46] that the form of the "extra" E 8 transformations suggests an interpretation in terms of a connection. We will build on the latter observation, and develop an E 8 geometry.…”

“…The occurrence of a two-derivative term points strongly to the transformation of a connection [46]. Let us perform a geometric check of the covariance (and, thereby, the closure), which we know will fail, but which will give interesting information.…”

The landscape of G-structures in eight-manifold compactifications of M-theory

“…This stems from difficulties 10 in building an appropriate generalized connection in eight dimensions, which in turn relates to the presence of Kaluza-Klein monopoles in the U-duality algebra and hence to the problem of including "dual gravitons" at the nonlinear level in E 8(8) -covariant formulations of eleven-dimensional supergravity [11][12][13][14] (which is obstructed by the no-go results of [15,16]). A solution to this problem was recently proposed in [51] within the framework of exceptional field theory but, as pointed out in [52], that solution may be incomplete. It would be interesting to understand what light may be shed on our results by exceptional generalized geometry.…”

Exceptional geometry and Borcherds superalgebras

“…Our results may lead to a way of including gravitational degrees of freedom in the Borcherds approach to supergravity, as well as to deeper insights into exceptional geometry. Since our results are generic for n ď 7 they may provide some guidance in dealing with the difficulties associated to the dual graviton in the case n " 8 [18,20,21], and in proceeding to n ě 9. Unless otherwise stated, 3 ď n ď 7 in this paper, but we will also comment on the case n " 8.…”

“…Considering B n and e n`1 as subalgebras of B n`1 we can generalize (2.14) to the set of relations 20) which will be useful in the next section.…”