Thermodynamics of noncritical M-theory and the topological A-model

In [P. Hořava, C.A. Keeler, Noncritical M-theory in 2 + 1 dimensions as a nonrelativistic Fermi liquid, hep-th/0508024], noncritical M-theory for two-dimensional type 0A and 0B strings was defined in terms of a double-scaled theory of nonrelativistic fermions in 2 + 1 dimensions. Here we study this noncritical M-theory at finite temperature. We derive the exact expression for the free energy of its vacuum solution, as a function of a coupling constant g_M and the radius R of the thermal circle. We show that at high temperature, the theory is effectively described by another M-theory solution, whose effective loop-counting coupling scales in a novel way characteristic of M-theory, as T³. Our calculations further suggest that noncritical M-theory is dual to the closed string theory of the topological A-model on a Calabi-Yau, with the radius R of the Euclidean time circle in M-theory playing the role of the string coupling constant of the A-model. In this correspondence, T-duality on the Euclidean time circle of noncritical M-theory implies an S-duality for the topological A-model.