Milan Niestijl

Positive Energy Representations of Gauge Groups With Support at a Fixed Point.

Let $\mathcal{K} \to M$ be a locally trivial smooth bundle of Lie groups equipped with an action of some Lie group $P$ by bundle automorphisms. Complementing recent progress B. Janssens and K.H. Neeb on the case where the $P$-action on $M$ has no fixed-points, projective unitary representations $\overline{\rho}$ of the locally-convex Lie group $\mathcal{G} := \Gamma_c(\mathcal{K})$ are studied which are of "positive energy" and factor entirely through the germs at some fixed point $a \in M$ of the $P$-action. Under suitable assumptions, it is shown that the kernel of a particular quadratic form on $\mathbb R[[x_1,\cdots, x_d]]$ generates an ideal in $\mathfrak{G} := \mathcal{Lie}(\mathcal{G})$ on which the derived representation $d\rho$ must vanish. This leads in particular to sufficient conditions for $d\rho$ to factor through a finite jet space $J^k_x(\mathcal{K})$ or through $J^\infty_a(N, \mathcal{K})$ for some usually lower-dimensional submanifold $N$. Some examples are considered. If time permits, we will have a closer look at the special case where $P=S^1$.

XXXIX Workshop on Geometric Methods in Physics	19.06–25.06.2022
XI School on Geometry and Physics	27.06–1.07.2022

Event sponsored by:
University of Bialystok