quantum gravity - What are the structures that exist at roughly the scale of the Planck length?

What are the mathematical/geometric structures that exist at the level of the Planck length?

Answer

"What is spacetime at Planck length? Once we understand that we will be able to formulate the theory of Quantum Gravity..." This is from Zizzy's Spacetime at the Planck Scale: The Quantum Computer View. So to get an answer one has to pick one of the current contenders, string theory, loop quantum gravity, or lesser known non-commutative geometry, causal set theory and cellular networks. In all of them "structure at the scale of Planck length" is not any kind of spatial/geometric structure, but rather something that we can only describe abstractly, or in the pretend spirit of "wave-particle duality". Ben Crowell's answer in Math Overflow thread describes the "minimal proposal", where one simply takes complex linear combinations of spacetimes. When I try to visualize that I see a bunch of shapes cutting through each other. But nevermind Planck scale, how does one even picture the superposition of dead and alive cat?

There is a Space at Planck Scale thread on Physics Overflow describing string theory picture, which includes variations on the minimum, only in 11 dimensions, those are non-linear sigma models. But in general "spacetime emerges" from some algebraic structures, which in the simplest (!) case are "Frobenius algebra objects internal to the modular representation categories of rational vertex operator algebras".

Zizzy himself advocates a version of LQG, where space-time is a superposition of "pixels" each of which encodes a qubit. More traditional LQG involves spin networks and spin foams, generally "spacetime is described as a quantum superposition of labelled piecewise-linear CW complexes or other related combinatorial/algebraic entities", see John Baez's references in the Math Overflow thread.