Description
The inverse problem of Scherk-Schwarz compactification of M-theory consists in distinguishing which lower dimensional maximal (gauged) supergravities admit a higher dimensional uplift, assuming the Scherk-Schwarz compactification to be the unique way to get back the initial theory. A general procedure is known in solving the uplift problem, for dimensions higher or equal to four, based on the generalised geometry formalism. The possibility of finding an uplift results in a condition the embedding tensor, which completely characterises the gauged supergravity, has to fulfill.
We will focus on the three-dimensional case. The main problem is that the fundamental representation of $E_{8(8)}$, the duality group of 3d maximal supergravities, is larger than the number of M-theory degrees of freedom, even including the dual graviton. To overcome the problem, one has to modify the generalised geometry, by ``doubling" the generalised Lie derivative. We will show how to extend the known procedure of finding an uplift in this modified generalised geometry, proving that also in this case a constraint has to be fulfilled by the embedding tensor in order an uplift to exist. We will also discuss the global patching of gauge parameters in this setup. [Based on 2409.xxxxx w/ G. Inverso]
