Matrix models for noncommutative algebraic manifolds

TL;DR

This paper introduces matrix models for algebraic submanifolds of the free complex sphere, explores their universal properties, and constructs higher half-liberations with faithful matrix models, advancing understanding of noncommutative algebraic manifolds.

Contribution

It develops a universal matrix model framework for algebraic submanifolds of the free complex sphere and constructs new higher half-liberations with faithful matrix models.

Findings

Existence of universal matrix models for fixed K

Inclusion chain of submanifolds $X^{(1)} o X^{(2)} o ...$

Construction of higher half-liberations with faithful models

Abstract

We discuss the notion of matrix model, $\pi:C(X)\to M_K(C(T))$, for algebraic submanifolds of the free complex sphere, $X\subset S^{N-1}_{\mathbb C,+}$. When $K\in\mathbb N$ is fixed there is a universal such model, which factorizes as $\pi:C(X)\to C(X^{(K)})\subset M_K(C(T))$. We have $X^{(1)}=X_{class}$ and, under a mild assumption, inclusions $X^{(1)}\subset X^{(2)}\subset X^{(3)}\subset\ldots\subset X$. Our main results concern $X^{(2)},X^{(3)},X^{(4)},\ldots$, their relation with various half-classical versions of $X$, and lead to the construction of families of higher half-liberations of the complex spheres and of the unitary groups, all having faithful matrix models.