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I have given the entry an Idea-section. Currently it reads as follows:
In the context of gauging of U-duality-symmetry groups of supergravity-theories to gauged supergravity, the embedding tensor (Nicolai-Samtleben 00) is the datum that specifies which subgroup of the global U-duality is promoted to a gauge group. The requirement of supersymmetry and of consistency then implies conditions on this choice, called the “linear constraint” and the “quadratic constraint”.
Formalized in terms of Lie theory (Lavau 17) these conditions say that an embedding tensor is a homomorphism of Leibniz algebras from a Lie module to the underlying Lie algebra (the “quadratic constaint”) where the Leibniz-product on the module is given by the Lie action induced by that homomorphism itself (the “linear constraint”).
Any choice of embedding tensor for a gauged supergravity is supposed to induce a tensor hierarchy (de Wit-Samtleben 05, 08) of higher rank tensor-fields which jointly serve as ever higher order corrections to the resulting gauge-covariance of the field strengths. This tensor hierarchy may be understood as a dg-Lie algebra/L-∞ algebra-structure which lifts the Leibniz algebra-structure implied/induced by the embedding tensor (Lavau 17, Lavau-Palmkvist 19, Lavau-Stasheff 19).
Added just a little pointer to the good review
{#HohmSamtleben19} Henning Samtleben, Olaf Hohm, Higher Gauge Structures in Double and Exceptional Field Theory (arXiv:1903.02821)
Also, I look forward to see how this topic will fit in the Hypothesis H program!
Thanks. Of course much more literature could still be added.
I find appreciable Lavau’s effort to clean up the topic of the tensor hierarchy, finally. Am trying to get to the bottom of it.
But with flavor branes in mind (and you might take this as the loose connection to Hypothesis H now) I thought it is interesting to note that an example of the partial gauging of a global symmetry group is the hidden local symmetry (that makes the chiral perturbation theory of pions accomodate vector mesons). The existing literature on flavor branes doesn’t address the subtlety involved in this step.
Added a pointer to phd thesis Edvard Musaev, $U-dualities in Type II string theories and M-theory (arXiv:1311.3331)
Added a pointer to phd thesis Edvard Musaev, $U-dualities in Type II string theories and M-theory (arXiv:1311.3331)
Thanks. Maybe this should (also) go to the entry U-duality?
BTW there is this published version spire:1749492 (at least: a published article with the same title)
You’re right, it’s a shorter published version of the original PhD thesis
Thanks for fixing my edit
The definition in question (here) is straight from the cited references, isn’t it. These references all follow the inductive style, declaring the underlying vector spaces to be:
$MultEnd(V)_1 \coloneqq V$
$MultEnd(V)_{n-1} \coloneqq Hom_k(V, MultEnd(V)_n)$.
All I added to this (in equation (1)) is the remark that hence
$MultEnd(V)_{-k} \simeq Hom_k( V^{\otimes_{k+1}}, V )$Also the bracket on these spaces is taken straight from the cited references. Seen under the above rewriting, the bracket between multilinear maps is, recursively, the multilinear map whose evaluation in the first argument on a vector $v$ is the bracket of multilinear maps with one of the two evaluated in their first argument on the given vector. This ought to have a Gerstenhaber-style formulation, true; probably it’s straightforward, but I didn’t pursue this and nothing in the entry depends on it.
Generally, there is nothing original in this entry, it’s just a summary of some concepts used in the literature. Citations are given.
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