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It should all be compatible in the end, keeping in mind the limits of available proposals.
For instance, we found that:
the Green-Schwarz action functional for the supermembrane (whose regularized quantization is the BFSS matrix model) is part of the rational equivariant Cohomotopy data postulated by Hypothesis H (this is Prop. 6.10 in Sec. 6.2 of arXiv:1805.05987);
the quantum ground states of the BMN version of the BFSS matrix model are plausibly seen among the quantum states of M-theory according to Hypothesis H – this is the content of Sec. 4.9 of arXiv:1912.10425,
I touch upon this in my recent talk Some Quantum States of M-Branes under Hypothesis H.
But one should note that there are still limitations on both sides:
(1) The BFSS matrix model is thought to see some corner of M-theory, but not the full thing (references here). For instance the BMN matrix model is a generalization/deformation of the BFSS model which clearly sees a different corner of M-theory.
(2) We haven’t really pushed for a full definition of M-theory via Hypothesis H just yet. Most of the checks we made so far concern differential-topological conditions of the flux fields over classical background spacetimes. From there we started exploring the quantum implications of the Hypothesis in arXiv:1912.10425 and arXiv:2105.02871, but this is not meant to be complete yet.
Notably, these discussions of Hypothesis H concern only the C-field so far, ignoring the field of (super-)gravity. It is hence clear that this cannot be the full picture yet, and that there is more detailed implications of Hypothesis H to be worked out. We think we know how to include the metric field (this is briefly indicated on slide 14 here) but analyzing this in detail remains to be done.
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