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We formulate a free probabilistic analog of the Wasserstein manifold on$\mathbb{R}^d$ (the formal Riemannian manifold of smooth probability densitieson $\mathbb{R}^d$), and we use it to study smooth non-commutative transport ofmeasure. The points of the free Wasserstein manifold$\mathscr{W}(\mathbb{R}^{*d})$ are smooth tracial non-commutative functions $V$with quadratic growth at $\infty$, which correspond to minus the log-density inthe classical setting. The space of smooth tracial non-commutative functionsused here is a new one whose definition and basic properties we develop in thepaper; they are scalar-valued functions of self-adjoint $d$-tuples fromarbitrary tracial von Neumann algebras that can be approximated by tracepolynomials. The space of non-commutative diffeomorphisms$\mathscr{D}(\mathbb{R}^{*d})$ acts on $\mathscr{W}(\mathbb{R}^{*d})$ bytransport, and the basic relationship between tangent vectors for$\mathscr{D}(\mathbb{R}^{*d})$ and tangent vectors for$\mathscr{W}(\mathbb{R}^{*d})$ is described using the Laplacian $L_V$associated to $V$ and its pseudo-inverse $\Psi_V$ (when defined). Followingsimilar arguments to arXiv:1204.2182, arXiv:1701.00132, and arXiv:1906.10051 inthe new setting, we give a rigorous proof for the existence of smooth transportalong any path $t \mapsto V_t$ when $V$ is sufficiently close $(1/2) \sum_j\operatorname{tr}(x_j^2)$, as well as smooth triangular transport.

Tracial smooth functions of non-commuting variables and the free Wasserstein manifold