Matrix models for noncommutative algebraic manifolds

Abstract

We discuss the notion of matrix model, $\pi :C(X)\to M_K(C(T))$, for algebraic submanifolds of the free complex sphere, $X\subset S_{\mathbb{C},+}^{N-1}$. 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 $X^{(1)}\subset X^{(2)}\subset X^{(3)}\subset \dots \subset X$. Our main results concern $X^{(2)},X^{(3)},X^{(4)},\dots$, 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.

25 pages. Final version

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