Hamiltonian spectral invariants, symplectic spinors and Frobenius structures I

Abstract

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 $\Phi$ 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 $\Phi$. We give lower bounds for the number of fixed points of $\Phi$ by defining a $C^*$-valued function on $T^*\tilde M$ defined by matrix coeficients of the Heisenberg group acting on spinors, where $\tilde M$ is a certain 'complexification' of $M$, whose critical points are in bijection to the fixed points of $\Phi$ resp. to the intersection of the spectral Lagrangian with the zero section $T^*\tilde M$. 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 $T^*\tilde M$ and using minimax-methods for 'half-infinite' chains.