2019/01/17 by Andreas Klein, Klein, Andreas
Mathematics · #Advanced Algebra and Geometry #Algebraic structures and combinatorial models #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Homotopy and Cohomology in Algebraic Topology #Mathematical Physics (math-ph) #Symplectic Geometry (math.SG)
paper · pdf · doi:10.48550/arxiv.1901.05605
openalex publication_date 2019/01/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 C1-small Hamiltonian mappings on symplectic manifolds M admitting a metaplectic structure and a parallel O(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 Φ 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 C1-small Hamiltonian mapping emerge, these dual pairs are essentially pairs (s1, J1), (s2, J2) of closed sections of the cotangent bundle T^*M and (in general singular) compatible 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 C1-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 (s1, J1), (s2, J2). We propose extensions of these techniques to the case of 'geodesic convexity-smallness' of Φ and to the case of general Hamiltonian systems on M.