Hamiltonian fixed points, symplectic spinors and Frobenius structures

Abstract

This article announces a series of articles aiming at introducing the concept of symplectic spinors into symplectic topology resp. the concept of Frobenius structures. We will give lower bounds for the number of fixed points of a Hamiltonian diffeomorphism on the cotangent bundle over a compact manifold $M$ by defining a certain $C^*$-valued function on $T^*\tilde M$, where $\tilde M$ is a certain 'complexification' of $M$, whose critical points are closely related to the fixed points of the Hamiltonian diffeomorphism $\Phi$ in question. This function, defined via embedding $\tilde M$ into $\mathbb{R}^m$ for an appopriate $m$ and the use of symplectic spinors, is essentially determined by associating to each point of $T^*\tilde M$ the value of a certain spinor-matrix coefficient of specific elements of the Heisenberg group which are determined by $\Phi$. We will discuss an approach for the case of the torus $M$ which does not require embeddings. Here, the matrix coefficients in question coincide with a certain theta function associated to the Hamiltonian diffeomorphism. We will 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. Furthermore we will define a 'Frobenius structure' on $T^*\tilde M$ by letting elements of $T(T^*\tilde M)$ act on the fibres of a line bundle $E$ on $T^*\tilde M$ spanned by 'coherent states' closely related to the above spinor-matrix coefficient. The spectral Lagrangian in $T^*(T^*\tilde M)$ associated to this Frobenius structure intersects the zero-section $T^*\tilde M$ exactly at the critical points of the function described beforehand.