Softcover ISBN:  9783985470679 
Product Code:  EMSMEM/9 
List Price:  $75.00 
AMS Member Price:  $60.00 
Softcover ISBN:  9783985470679 
Product Code:  EMSMEM/9 
List Price:  $75.00 
AMS Member Price:  $60.00 

Book DetailsMemoirs of the European Mathematical SocietyVolume: 9; 2024; 156 ppMSC: Primary 22; Secondary 17
This is the first in a series of papers on projective positive energy representations of gauge groups. Let \(\Xi \rightarrow M\) be a principal fiber bundle, and let \(\Gamma_{C}(M,\mathrm{Ad}(\Xi))\) be the group of compactly supported (local) gauge transformations. If \(P\) is a group of “spacetime symmetries” acting on \(\Xi \rightarrow M\), then a projective unitary representation of \(\Gamma_{C}(M,\mathrm{Ad}(\Xi))\rtimes P\) is of positive energy if every “timelike generator” \(p_{0} \in p\) gives rise to a Hamiltonian \(H(p_{0})\) whose spectrum is bounded from below. The authors' 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 \( \Gamma_{C}(M,\textrm{AD}(\Xi))_{0}\) come from 1dimensional \(P\)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, the authors consider larger groups of gauge transformations, which also contain global transformations. The present results are used to localize the positive energy representations at (conformal) infinity.
A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society.

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This is the first in a series of papers on projective positive energy representations of gauge groups. Let \(\Xi \rightarrow M\) be a principal fiber bundle, and let \(\Gamma_{C}(M,\mathrm{Ad}(\Xi))\) be the group of compactly supported (local) gauge transformations. If \(P\) is a group of “spacetime symmetries” acting on \(\Xi \rightarrow M\), then a projective unitary representation of \(\Gamma_{C}(M,\mathrm{Ad}(\Xi))\rtimes P\) is of positive energy if every “timelike generator” \(p_{0} \in p\) gives rise to a Hamiltonian \(H(p_{0})\) whose spectrum is bounded from below. The authors' 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 \( \Gamma_{C}(M,\textrm{AD}(\Xi))_{0}\) come from 1dimensional \(P\)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, the authors consider larger groups of gauge transformations, which also contain global transformations. The present results are used to localize the positive energy representations at (conformal) infinity.
A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society.