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In this article, we continue our study of 'Frobenius structures' andsymplectic spectral invariants in the context of symplectic spinors. Bystudying the case of $C^1$-small Hamiltonian mappings on symplectic manifolds$M$ admitting a metaplectic structure and a parallel $\hat O(n)$-reduction ofits metaplectic frame bundle we derive how the construction of 'singularlyrigid' resp. 'self-dual' pairs of irreducible Frobenius structures associatedto this Hamiltonian mapping $\Phi$ leads to a Hopf-algebra-type structure onthe set of irreducible Frobenius structures. We then generalize thisconstruction and define abstractly conditions under which 'dual pairs'associated to a given $C^1$-small Hamiltonian mapping emerge, these dual pairsare essentially pairs $(s_1, J_1), (s_2, J_2)$ of closed sections of thecotangent bundle $T^*M$ and (in general singular) compatible almost complexstructures on $M$ satisfying certain integrability conditions involving aKoszul bracket. In the second part of this paper, we translate thesecharacterizing conditions for general 'dual pairs' of Frobenius structuresassociated to a $C^1$-small Hamiltonian system into the notion of matrixfactorization. We propose an algebraic setting involving modules over certainfractional ideals of function rings on $M$ so that the set of 'dual pairs' inthe above sense and the set of matrix factorizations associated to thesemodules stand in bijective relation. We prove, in the real-analytic case, aRiemann Roch-type theorem relating a certain Euler characteristic arising froma given matrix factorization in the above sense to (integral) cohomologicaldata on $M$ using Cheeger-Simons-type differential characters, derived from agiven pair $(s_1, J_1), (s_2, J_2)$. We propose extensions of these techniquesto the case of 'geodesic convexity-smallness' of $\Phi$ and to the case ofgeneral Hamiltonian systems on $M$.

Hamiltonian spectral invariants, symplectic spinors and Frobenius structures II