N = 4: a unifying framework for 2d topological gravity, cM ≤ 1 string theory and constrained topological sigma model

It is shown that two-dimensional (2d) topological gravity in the conformal gauge has a larger symmetry than has been hitherto recognized; in the formulation of Labastida, Pernici and Witten it contains a twisted “small” N = 4 superconformal symmetry. There are in fact two distinct twisted N = 2 structures within this N = 4, one of which is shown to be isomorphic to the algebra discussed by the Verlindes and the other corresponds, through bosonization, to cM ≤ 1 string theory discussed by Bershadsky et al. As a byproduct, we find a twisted N = 4 structure in cM ≤ 1 string theory. We also study the “mirror” of this twisted N = 4 algebra and find that it corresponds, through another bosonization, to a constrained topological sigma model in complex dimension one.