  Inspired by the low wave-length limit of topological M-theory, which
re-constructs the theory of $3+1$D gravity in the self-dual variables'
formulation, and by the realization that in Loop Quantum Gravity the holonomy
of a flat connection can be non-trivial if and only if a non-trivial
(space-like) line defect is localized inside the loop, we argue that
non-trivial gravitational holonomies can be put in correspondence with
space-like M-branes. This suggests the existence of a new duality, which we
call $H$ duality, interconnecting topological M-theory with Loop Quantum
Gravity. We spell some arguments to show that fundamental S-strings are serious
candidates to be considered in order to instantiate this correspondence to
classes of LQG states. In particular, we consider the case of the holonomy
flowers in LQG, and show that for this type of states the action of the
Hamiltonian constraint, from the M-theory side, corresponds to a linear
combination of appearance and disappearance of a SNS1- strings. Consequently,
these processes can be reinterpreted, respectively, as enucleations or decays
into open or closed strings.
