[Paper Review] Positive energy representations of gauge groups I: Localization

Bas Janssens, Karl‐Hermann Neeb|arXiv (Cornell University)|Aug 7, 2021

Black Holes and Theoretical Physics59 references4 citations

TL;DR

This paper establishes a localization theorem for projective positive energy representations of compactly supported gauge groups $Γ_c(M,\operatorname{Ad}(\Xi))$ associated with a principal bundle $\Xi \to M$. It shows that, under the absence of fixed points for the cone of timelike generators, such representations arise from 1-dimensional $P$-invariant submanifolds $S \subseteq M$, leading to a complete classification in terms of lowest weight representations of affine Kac–Moody algebras when $M$ is compact.

ABSTRACT

This is the first in a series of papers on projective positive energy representations of gauge groups. Let $Ξ ightarrow M$ be a principal fiber bundle, and let $Γ_{c}(M,\mathrm{Ad}(Ξ))$ be the group of compactly supported (local) gauge transformations. If $P$ is a group of `space-time symmetries' acting on $Ξ ightarrow M$, then a projective unitary representation of $Γ_{c}(M,\mathrm{Ad}(Ξ)) times P$ is of positive energy if every `timelike generator' $p_0 \in \mathfrak{p}$ gives rise to a Hamiltonian $H(p_0)$ whose spectrum is bounded from below. Our main result shows that in the absence of fixed points for the cone of timelike generators, the projective positive energy representations of the connected component $Γ_{c}(M,\mathrm{Ad}(Ξ))_0$ come from 1-dimensional $P$-orbits. For compact $M$ this yields a complete classification of the projective positive energy representations in terms of lowest weight representations of affine Kac-Moody algebras. For noncompact $M$, it yields a classification under further restrictions on the space of ground states. In the second part of this series we consider larger groups of gauge transformations, which contain also global transformations. The present results are used to localize the positive energy representations at (conformal) infinity.

Motivation & Objective

• To extend the Wigner–Mackey classification of positive energy representations to infinite-dimensional gauge groups by unifying internal and space-time symmetries.
• To classify projective unitary positive energy representations of the group $\Gamma_c(M,\operatorname{Ad}(\Xi)) \rtimes P$, where $P$ acts as space-time symmetries on a principal bundle $\Xi \to M$.
• To establish a localization mechanism that reduces representations on $M$ to representations on 1-dimensional $P$-invariant submanifolds $S \subseteq M$.
• To provide a complete classification of such representations when $M$ is compact, in terms of lowest weight representations of affine Kac–Moody algebras.
• To lay the foundation for Part II, which extends the analysis to global gauge transformations and their localization at conformal infinity.

Proposed method

• Analyzes projective unitary representations of the semidirect product $\Gamma_c(M,\operatorname{Ad}(\Xi)) \rtimes P$, where $P$ is a group of space-time symmetries with a cone $\mathscr{C} \subseteq \mathfrak{p}$ of timelike generators.
• Applies the concept of positive energy by requiring that the Hamiltonian $H(p_0)$ associated to each $p_0 \in \mathscr{C}$ has a spectrum bounded from below.
• Uses Lie algebra techniques to reduce the problem to the study of local gauge algebras and their central extensions, particularly focusing on twisted loop algebras.
• Employs Sobolev–Lie algebra structures and continuous extension theorems to extend representations from smooth to Sobolev spaces.
• Applies a localization theorem at the Lie algebra level, showing that positive energy representations of $\Gamma_c(M,\operatorname{Ad}(\Xi))_0$ factor through representations on 1-dimensional $P$-orbits.
• Utilizes ergodicity properties of 1-parameter subgroups in $\widetilde{\mathrm{SL}}(2,\mathbb{R})$ to prove that vacuum vectors are fixed under full group actions, enabling classification.

Experimental results

Research questions

• RQ1 Under what conditions do projective positive energy representations of $\Gamma_c(M,\operatorname{Ad}(\Xi))_0$ localize to 1-dimensional $P$-invariant submanifolds?
• RQ2 How can positive energy representations of gauge groups be classified when the base manifold $M$ is compact?
• RQ3 What role do central extensions and Sobolev spaces play in the continuous extension of positive energy representations?
• RQ4 How does the absence of fixed points for the timelike generator cone affect the structure of positive energy representations?
• RQ5 What is the relationship between vacuum representations and the ergodic properties of 1-parameter subgroups in $\widetilde{\mathrm{SL}}(2,\mathbb{R})$?

Key findings

• The localization theorem establishes that every projective positive energy representation of $\Gamma_c(M,\operatorname{Ad}(\Xi))_0$ arises from a 1-dimensional $P$-invariant submanifold $S \subseteq M$.
• For compact $M$, the classification of projective positive energy representations is complete and equivalent to lowest weight representations of affine Kac–Moody algebras.
• The representations extend continuously to Sobolev spaces, ensuring smoothness and compatibility with the box manifold structure.
• The vacuum vector $\Omega$ is fixed by the entire group $\widetilde{\mathrm{SL}}(2,\mathbb{R})$ if it is fixed by any 1-parameter subgroup $\widetilde{x}(t)$ or $\widetilde{h}(t)$, implying full invariance.
• The continuous extension theorem ensures that smoothness of the representation map $\varphi = \Phi_3 \circ \Phi_2 \circ \Phi_1$ follows from the smoothness of $\Phi_1$, continuity of $\Phi_2$, and holomorphicity of $\Phi_3$.
• The classification for noncompact $M$ is obtained under additional restrictions on the space of ground states, indicating a more constrained structure compared to the compact case.