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Equivalent Laplacian in non-commutative geometry in a general context?

I’m working on the problem of finding nontrivial solutions to the heat equation in dimension $n$. The problem is, in the commutative setting, that non trivial solutions do not exist by the Liouville theorem.
It has always been my experience that non-commutative theories are “rough”. I’ve never worked with them in detail, but I imagine that the non-commutative theories have some features similar to what you have in general relativity. For instance, if we would like to work in a non-commutative theory with the commutator given by $[\hat{X}_i,\hat{X}_j]=i \theta_{ij}$ with $\theta_{ij}$ some constant skewsymmetric tensor, is it possible to have a Laplacian whose kernel will contain nontrivial solutions to the heat equation?
In the specific case of general relativity, as people know, it is possible to have nontrivial perturbations on the geometry of space-time. What are the equivalent mathematical conditions on $\theta_{ij}$ so that the non-commutative theory we’re working with produces nontrivial perturbations for the metric?
Or maybe we can have a “singular” Laplacian on the non-commutative space whose kernel will have trivial solutions?
In particular I’m interested in the case of $3+1$ dimensions.
Thanks!

A:

The Laplacian of a metric $g$ of a $4$-d manifold can be defined on spinors (cf. P. G. Bergman’s metric spinors, in J. R. Isenberg, P. M. Kunst, A. S. Cattaneo, eds, “Symplectic Geometry, Groupoids, and Integrable Systems.” Contemporary Math. vol. 201, 1999, AMS, Providence, RI, 52–55) by
 \Delta_{g} s = – D_{g}D_{g}s
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