Free noncommutative principal divisors and commutativity of the tracial fundamental group

TL;DR

This paper introduces a theory of principal divisors for free noncommutative functions, compares their singularity sets, and explores the structure of fundamental groups, revealing they are abelian and composed of rational numbers.

Contribution

It defines principal divisors for free noncommutative functions and develops a cohomology and fundamental group theory for tracial free functions, highlighting their abelian nature.

Findings

Divisors of noncommutative rational functions are differences of polynomial divisors.

The fundamental group for tracial free functions is a direct sum of copies of .

Contrasts classical and free cases, showing abelian and rational structure.

Abstract

We define the principal divisor of a free noncommuatative function. We use these divisors to compare the determinantal singularity sets of free noncommutative functions. We show that the divisor of a noncommutative rational function is the difference of two polynomial divisors. We formulate a nontrivial theory of cohomology, fundamental groups and covering spaces for tracial free functions. We show that the natural fundamental group arising from analytic continuation for tracial free functions is a direct sum of copies of $\mathbb{Q}$. Our results contrast the classical case, where the analogous groups may not be abelian, and the free case, where free universal monodromy implies such notions would be trivial.