W∞ and anomalies of self-dual Einstein theories

Recently it has been demonstrated that self-dual Einstein Euclidean instantons possess an infinite-dimensional group of symmetries which contain the standard ω∞. This paper reports that, since ω∞ has a central extension only in its ω2 subalgebra one may claim that there is an anomaly in the symmetry group of these instantons which is described by a 3-dimensional ω2 effective action. Thus one has a bosonization of the 3/2 spin anomaly of the Eguchi-Hanson instantons. In analogy to ω∞ we construct the 3-dimensional effective action for the Lone Star W∞ algebra. This suggests that there is a quantum deformation of the instantons that contributes to the effective action of 4-dimensional self-dual gravity. The authors comment on this possibility