Formal Groups, Witt vectors and Free Probability

Abstract

We establish a link between free probability theory and Witt vectors, via the theory of formal groups. We derive an exponential isomorphism which expresses Voiculescu's free multiplicative convolution $\boxtimes$ as a function of the free additive convolution $\boxplus$. Subsequently we continue our previous discussion of the relation between complex cobordism and free probability. We show that the generic $n$th free cumulant corresponds to the cobordism class of the $(n-1)$-dimensional complex projective space. This permits us to relate several probability distributions from random matrix theory to known genera, and to build a dictionary. Finally, we discuss aspects of free probability and the asymptotic representation theory of the symmetric group from a conformal field theoretic perspective and show that every distribution with mean zero is embeddable into the Universal Grassmannian of Sato-Segal-Wilson.