2014/11/16 by Andreas Klein, Klein, Andreas
Biochemistry, Genetics and Molecular Biology · Mathematics · Physics and Astronomy · #53D05 #53D37 #53D45 #Advanced Algebra and Geometry #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Homotopy and Cohomology in Algebraic Topology #Mathematical Physics (math-ph) #Microtubule and mitosis dynamics #Nonlinear Waves and Solitons #Symplectic Geometry (math.SG) #math-ph #math.DG #math.MP #math.SG #msc:53D05 #msc:53D37 #msc:53D45
paper · pdf · doi:10.48550/arxiv.1411.4237
47 pages, v6: added a discussion of the indecomposable case, v7: Proposition 3.13 and dependent theorems rewritten, minor further corrections
openalex publication_date 2014/11/16 · arxiv created 2016/01/24 · arxiv updated 2016/01/26 · openalex created_date 2022/10/03 · openalex updated_date 2026/07/28
This is the first of two articles aiming to introduce symplectic spinors into the field of symplectic topology and the subject of Frobenius structures. After exhibiting a (tentative) axiomating setting for Frobenius structures resp. 'Higgs pairs' in the context of symplectic spinors, we present immediate observations concerning a local Schroedinger equation, the first structure connection and the existence of 'spectrum', its topological interpretation and its connection to 'formality' which are valid for the case of standard Frobenius structures. We give a classification of the irreducibles and the indecomposables of the latter in terms of certain U(n)-reductions of the G-extension of the metaplectic frame bundle and a certain connection on it, where G is the semi-direct product of the metaplectic group and the Heisenberg group, while the indecomposable case involves in addition the combinatorial structure of the eigenstates of the n-dimensional harmonic oscillator. In the second part, we associate an irreducible Frobenius structure to any Hamiltonian diffeomorphism Φ on a cotangent bundle T^*M. The spectral Lagrangian in T^*(T^*M) associated to this Frobenius structure intersects the zero-section T^*M exactly at the fixed points of Φ. We give lower bounds for the number of fixed points of Φ by defining a C^*-valued function on T^* M defined by matrix coeficients of the Heisenberg group acting on spinors, where M is a certain 'complexification' of M, whose critical points are in bijection to the fixed points of Φ resp. to the intersection of the spectral Lagrangian with the zero section T^* M. We discuss how to define spectral invariants in the sense of Viterbo and Oh by lifting the above function to a real-valued function on an appropriate cyclic covering of T^* M and using minimax-methods for 'half-infinite' chains.