Generalized Positive Energy Representations of Groups of Jets

Abstract

Let $V$ be a finite-dimensional real vector space and $K$ a compact simple Lie group with Lie algebra $\mathfrak{k}$. Consider the Fr\'echet-Lie group $G := J_0^\infty(V; K)$ of $\infty$-jets at $0 \in V$ of smooth maps $V \to K$, with Lie algebra $\mathfrak{g} = J_0^\infty(V; \mathfrak{k})$. Let $P$ be a Lie group and write $\mathfrak{p} := \textrm{Lie}(P)$. Let $\alpha$ be a smooth $P$-action on $G$. We study smooth projective unitary representations $\bar{\rho}$ of $G \rtimes_\alpha P$ that satisfy a so-called generalized positive energy condition. In particular, this class captures representations that are in a suitable sense compatible with a KMS state on the von Neumann algebra generated by $\bar{\rho}(G)$. We show that this condition imposes severe restrictions on the derived representation $d\bar{\rho}$ of $\mathfrak{g} \rtimes \mathfrak{p}$, leading in particular to sufficient conditions for $\bar{\rho}\big|_{G}$ to factor through $J_0^2(V; K)$, or even through $K$.