$W_\infty$ 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 {omega}{sub {infinity}}. This paper reports that, since {omega}{sub {infinity}} has a central extension only in its {omega}{sub 2} subalgebra one may claim that there is an anomaly in the symmetry group of these instantons which is described by a 3-dimensional {omega}{sub 2} effective action. Thus one has a bosonization of the 3/2 spin anomaly of the Eguchi-Hanson instantons. In analogy to {omega}{sub {infinity}} we construct the 3-dimensional effective action for the Lone Star W{sub {infinity}} 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.