Projective families of Dirac operators on a Banach Lie groupoid

Abstract

We introduce a Banach Lie group $G$ of unitary operators subject to a natural trace condition. We compute the homotopy groups of $G$, describe its cohomology and construct an $S^1$-central extension. We show that the central extension determines a non-trivial gerbe on the action Lie groupoid $G⋉\mathfrak{k}$, where $\mathfrak{k}$ denotes the Hilbert space of self-adjoint Hilbert–Schmidt operators. With an eye towards constructing elements in twisted K-theory, we prove the existence of a cubic Dirac operator $\mathbb{D}$ in a suitable completion of the quantum Weil algebra $\mathcal{U}(\mathfrak{g})\otimes Cl(\mathfrak{k})$, which is subsequently extended to a projective family of self-adjoint operators ${\mathbb{D}}_A$ on $G⋉\mathfrak{k}$. While the kernel of ${\mathbb{D}}_A$ is infinite-dimensional, we show that there is still a notion of finite reducibility at every point, which suggests a generalized definition of twisted K-theory for action Lie groupoids.