1306.2077 (Andreas Klein)

Hamiltonian fixed points, symplectic spinors and Frobenius structures    [PDF]

Andreas Klein

In this article (this is a research announcement), 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.

View original: http://arxiv.org/abs/1306.2077