There is a very beautiful story on several families of stable classes of the compactification of the moduli space of algebraic genus curves with marked points, playing major role in geometry (especially intersection theory), Gromov-Witten theory, etc. They are sometimes referred by standard notation and sometimes by their names; one has families of Mumford-Miller classes and of Morita classes.
Kontsevich associates to any cyclic/symplectic -algebra certain partition function which is a inhomogeneous class in graph homology. Igusa and Mondello have substantiated Kontsevich’s claim that the Mumford-Miller-Morita classes are induced from certain family of -algebras; Hamilton-Lazarev have shown that the recipe for the classes from the -algebra is a homotopy invariant and that the partition function takes values in the relative homology of an infinite-dimensional algebra of noncommutative Hamiltonians. They interpret the class as being somewhat analogous to Chern classes and call it a characteristic class of the -algebra.
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