Three papers discussed in a recent thread at the Physics Forum discuss this possibility that String Theory and Loop Quantum Gravity, the two main approaches to quantum gravity theory, are equivalent. The papers are (emphasis and paragraph breaks in the abstracts are mine):
A new duality between Topological M-theory and Loop Quantum Gravity Andrea Addazi, Antonino Marciano (Submitted on 17 Jul 2017)
Inspired by the long wave-length limit of topological M-theory, which re-constructs the theory of 3+1D gravity in the self-dual variables' formulation, we conjecture the existence of a duality between Hilbert spaces, the H-duality, to unify topological M-theory and loop quantum gravity (LQG). By H-duality non-trivial gravitational holonomies of the kinematical Hilbert space of LQG correspond to space-like M-branes. The spinfoam approach captures the non-perturbative dynamics of space-like M-branes, and can be claimed to be dual to the S-branes foam. The Hamiltonian constraint dealt with in LQG is reinterpreted as a quantum superposition of SM-brane nucleations and decays.Cite as: arXiv:1707.05347 [hep-th]
New Variables for Classical and Quantum Gravity in all Dimensions I. Hamiltonian Analysis Norbert Bodendorfer, Thomas Thiemann, Andreas Thurn (Submitted on 18 May 2011 (v1), last revised 12 Feb 2013 (this version, v2))
Loop Quantum Gravity heavily relies on a connection formulation of General Relativity such that 1. the connection Poisson commutes with itself and 2. the corresponding gauge group is compact. This can be achieved starting from the Palatini or Holst action when imposing the time gauge. Unfortunately, this method is restricted to D+1 = 4 spacetime dimensions. However, interesting String theories and Supergravity theories require higher dimensions and it would therefore be desirable to have higher dimensional Supergravity loop quantisations at one's disposal in order to compare these approaches.
In this series of papers, we take first steps towards this goal. The present first paper develops a classical canonical platform for a higher dimensional connection formulation of the purely gravitational sector. The new ingredient is a different extension of the ADM phase space than the one used in LQG, which does not require the time gauge and which generalises to any dimension D > 1. The result is a Yang-Mills theory phase space subject to Gauss, spatial diffeomorphism and Hamiltonian constraint as well as one additional constraint, called the simplicity constraint. The structure group can be chosen to be SO(1,D) or SO(D+1) and the latter choice is preferred for purposes of quantisation.
Journal reference: Class. Quantum Grav. 30 (2013) 045001Cite as: arXiv:1105.3703 [gr-qc]
Should nature be supersymmetric, then it will be described by Quantum Supergravity at least in some energy regimes.
The currently most advanced description of Quantum Supergravity and beyond is Superstring Theory/M-Theory in 10/11 dimensions. String Theory is a top-to-bottom approach to Quantum Supergravity in that it postulates a new object, the string, from which classical Supergravity emerges as a low energy limit. On the other hand, one may try more traditional bottom-to-top routes and apply the techniques of Quantum Field Theory.
Loop Quantum Gravity (LQG) is a manifestly background independent and non-perturbative approach to the quantisation of classical General Relativity, however, so far mostly without supersymmetry.
Of course, it makes sense that any two quantum gravity theories that by design are supposed to approximate General Relativity in the classical limit, should have similarities to each other.The main obstacle to the extension of the techniques of LQG to the quantisation of higher dimensional Supergravity is that LQG rests on a specific connection formulation of General Relativity which exists only in D+1 = 4 dimensions. In this Letter we introduce a new connection formulation of General Relativity which exists in all space-time dimensions. We show that all LQG techniques developed in D+1 = 4 can be transferred to the new variables in all dimensions and describe how they can be generalised to the new types of fields that appear in Supergravity theories as compared to standard matter, specifically Rarita-Schwinger and p-form gauge fields.Journal reference: Phys. Lett. B 711: 205-211 (2012)Cite as: arXiv:1106.1103 [gr-qc]