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 no
n-trivial gerbe on the action Lie groupoid $Gltimes mathfrak{k}$, 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 $Gltimes frak{k}$. 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.
We study the structure of abelian extensions of the group $L_qG$ of $q$-differentiable loops (in the Sobolev sense), generalizing from the case of central extension of the smooth loop group. This is motivated by the aim of understanding the problems
with current algebras in higher dimensions. Highest weight modules are constructed for the Lie algebra. The construction is extended to the current algebra of supersymmetric Wess-Zumino-Witten model. An application to the twisted K-theory on $G$ is discussed.
