Hamiltonian spectral invariants, symplectic spinors and Frobenius structures II

arXiv1901.05605MaRDI QIDQ6312581

Andreas Klein

Publication date: 16 January 2019

Abstract: In this article, we continue our study of 'Frobenius structures' and symplectic spectral invariants in the context of symplectic spinors. By studying the case of $C^1$-small Hamiltonian mappings on symplectic manifolds $M$ admitting a metaplectic structure and a parallel $hatO(n)$-reduction of its metaplectic frame bundle we derive how the construction of 'singularly rigid' resp. 'self-dual' pairs of irreducible Frobenius structures associated to this Hamiltonian mapping $Phi$ leads to a Hopf-algebra-type structure on the set of irreducible Frobenius structures. We then generalize this construction and define abstractly conditions under which 'dual pairs' associated to a given $C^1$-small Hamiltonian mapping emerge, these dual pairs are esssentially pairs $(s_1,J_1),(s_2,J_2)$ of closed sections of the cotangent bundle $T^{*}M$ and (in general singular) comptaible almost complex structures on $M$ satisfying certain integrability conditions involving a Koszul bracket. In the second part of this paper, we translate these characterizing conditions for general 'dual pairs' of Frobenius structures associated to a $C^1$-small Hamiltonian system into the notion of matrix factorization. We propose an algebraic setting involving modules over certain fractional ideals of function rings on $M$ so that the set of 'dual pairs' in the above sense and the set of matrix factorizations associated to these modules stand in bijective relation. We prove, in the real-analytic case, a Riemann Roch-type theorem relating a certain Euler characteristic arising from a given matrix factorization in the above sense to (integral) cohomological data on $M$ using Cheeger-Simons-type differential characters, derived from a given pair $(s_1,J_1),(s_2,J_2)$. We propose extensions of these techniques to the case of 'geodesic convexity-smallness' of $Phi$ and to the case of general Hamiltonian systems on $M$.

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