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 $⊠$ as a function of the free additive convolution $⊞$. 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.

Revised and substantially extended version. Contains an additional section on conformal field theory and free probability with new results. 31 pages with 1 figure

References in corpus (5)

• Second Order Freeness and Fluctuations of Random Matrices: II. Unitary Random Matrices
• Kondo effect in single atom contacts: the importance of the atomic geometry
• Hopf algebras and the logarithm of the S-transform in free probability
• Almost Commutative Probability Theory
• The $S$-transform in arbitrary dimensions

Cited by in corpus (7)

• The splitting process in free probability theory
• Almost Commutative Probability Theory
• Shuffle group laws. Applications in free probability
• (Co)monads in Free Probability Theory
• The $S$-transform in arbitrary dimensions
• Topological invariants of some chemical reaction networks
• Renormalization groupoids in algebraic topology