Abstract
We develop the generalized Cartan Calculus for the groups $$ G=\mathrm{S}\mathrm{L}\left(2,\mathrm{\mathbb{R}}\right)\times {\mathrm{\mathbb{R}}}^{+},\mathrm{S}\mathrm{L}\left(5,\mathrm{\mathbb{R}}\right) $$ and SO(5, 5). They are the underlying algebraic structures of d = 9, 7, 6 exceptional field theory, respectively. These algebraic identities are needed for the “tensor hierarchy” structure in exceptional field theory. The validity of Poincare lemmas in this new differential geometry is also discussed. Finally we explore some possible extension of the generalized Cartan calculus beyond the exceptional series.
Highlights
Remaining dimensions along with the exterior dimensions exactly give a 11-dimensional Mtheory solution or a 10-dimensional IIB supergravity solution
We develop the generalized Cartan Calculus for the groups G = SL(2, R) × R+, SL(5, R) and SO(5, 5)
A direct consequence of the modification of differential geometry is that the 2-form field strength associated to this 1-form gauge field is no longer gauge covariant
Summary
To construct a generalized Cartan Calculus, one first picks a symmetry group G, and a representation R of G in which the vector field V M lives. The ordinary (interior) Lie derivative of a vector, LΛV M = ΛN ∂N V M − V N ∂N ΛM ,. In this paper all the Z-tensors are symmetric in M N and P Q: ZMN P Q = ZNM P Q = ZMN QP = ZNM QP With this Z-tensor, the generalized Lie derivative can be rewritten in the following form: LΛV M = ΛN ∂N V M − V N ∂N ΛM + ZMN P Q∂N ΛP V Q + (λ − ω)∂N ΛN V M. In the following discussions all the gauge parameters Λ in generalized Lie derivative LΛ are assumed to have weight ω. For 5 ≤ d ≤ 8, α and the expression of ZMN P Q can be found in [18]
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