A candidate for a background independent formulation of M theory

TL;DR

Smolin advances a background independent formulation of ${\cal M}$ theory by modeling membranes via conformal blocks of a quantum group ${G_q}$ on all finite-genus two-surfaces. The bulk dynamics are governed by causal histories, while holographic, finite-dimensional boundary observables encode the interior through a loop/Chern-Simons map, naturally yielding matrix-model-like coordinates for D0-branes in suitable dimensions. In particular, the paper shows explicit constructions in $D=3$ and $D=9$ where the boundary dynamics reproduce bosonic and supersymmetric matrix models in appropriate limits, suggesting a continuum limit in which strings and D0-branes emerge from background independent membranes and linking these to M-theory, AdS/CFT, and black-hole entropy. Three central conjectures articulate how such a background independent theory could realize flat 11D spacetime, embed into a covariant ${Osp(1|32)}$ framework, and encompass perturbative string theories as different phases. The work outlines concrete steps to extend to higher dimensions, fermionic sectors, and brane dynamics, paving a path toward a non-perturbative, background independent realization of M theory via holographic matrix-model observables.

Abstract

A class of background independent membrane field theories are studied, and several properties are discovered which suggest that they may play a role in a background independent form of M theory. The bulk kinematics of these theories are described in terms of the conformal blocks of an algebra G on all oriented, finite genus, two-surfaces. The bulk dynamics is described in terms of causal histories in which time evolution is specified by giving amplitudes to certain local changes of the states. Holographic observables are defined which live in finite dimensional states spaces associated with boundaries in spacetime. We show here that the natural observables in these boundary state spaces are, when G is chosen to be Spin(D) or a supersymmetric extension of it, generalizations of matrix model coordinates in D dimensions. In certain cases the bulk dynamics can be chosen so the matrix model dynamics is recoverd for the boundary observables. The bosonic and supersymmetric cases in D=3 and D=9 are studied, and it is shown that the latter is, in a certain limit, related to the matrix model formulation of M theory. This correspondence gives rise to a conjecture concerning a background independent form of M theory in terms of which excitations of the background independent membrane field theory that correspond to strings and D0 branes are identified.

Figures (21)

• Figure 1: A trinion which represents the intertwiner of three representations of $G_{q}$.
• Figure 2: A simple example of two surface, and its dual graph $\Gamma$.
• Figure 3: A simple example of two surface, divided into four punctured spheres. The dual $3$-manifold is the boundary of the four-simplex.
• Figure 4: A quantum geometry $|\Gamma, j_a, \mu_{i}>$ is cut along $5$ circles, yielding two bounded quantum geometries, ${\cal S}_{g}^{\pm}$, which meet on a boundary which is represented by a $5$-punctured $S^{2}$.
• Figure 5: An example of a substitution move. Histories $\cal M$ are made up of successive applications of such moves.