They define the non-critical M-theory as a double scaling limit of non-relativistic fermions in 2+1 dimensions much like the usual M-theory may be defined using the large N limit of a non-relativistic description of D0-branes in type IIA. Petr and Cynthia only have the description in terms of fermions, not the matrix model itself. The off-diagonal elements of such a hypothetical matrix model should be pure gauge anyway, I think. They also show that noncritical type 0A and 0B theories appear as "hydrodynamic" solutions of their noncritical M-theory.
You may think that this work is another variation on the topic of extending the well-known dualities to uncontrollable non-supersymmetric cases. But there's a difference: in two spacetime dimensions, string theory becomes stable and the "tachyon" becomes a misnomer: the stringy "tachyon" actually becomes massless. The same thing morally holds for their non-critical M-theory in 2+1 dimensions. Because of this special feature, the system is actually exactly solvable - unlike the duality between type 0A in 10 dimensions and M-theory on the Scherk-Schwarz circle, for example. A general comment about this situation is that we know two important classes of exactly solvable systems, namely those with
- large enough spacetime supersymmetry algebras
- small enough spacetime dimensionalities
Even if these subcritical vacua are part of the full "generalized string/M-theory", it is less clear how they may be relevant for "the" string/M-theory that should explain the real world.
External links: Jacques Distler has a complementary description of the article with some math in it.