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Positive Energy Representations of Gauge Groups I: Localization
 
Bas Janssens Delft University of Technology, The Netherlands
Karl-Hermann Neeb Friedrich-Alexander University, Erlangen-Nuremberg, Germany
A publication of European Mathematical Society
Softcover ISBN:  978-3-98547-067-9
Product Code:  EMSMEM/9
List Price: $75.00
AMS Member Price: $60.00
Not yet published - Preorder Now!
Expected availability date: May 06, 2024
Please note AMS points can not be used for this product
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Positive Energy Representations of Gauge Groups I: Localization
Bas Janssens Delft University of Technology, The Netherlands
Karl-Hermann Neeb Friedrich-Alexander University, Erlangen-Nuremberg, Germany
A publication of European Mathematical Society
Softcover ISBN:  978-3-98547-067-9
Product Code:  EMSMEM/9
List Price: $75.00
AMS Member Price: $60.00
Not yet published - Preorder Now!
Expected availability date: May 06, 2024
Please note AMS points can not be used for this product
  • Book Details
     
     
    Memoirs of the European Mathematical Society
    Volume: 92024; 156 pp
    MSC: 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 “space-time 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 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, 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.

  • Additional Material
     
     
  • Requests
     
     
    Review Copy – for publishers of book reviews
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Volume: 92024; 156 pp
MSC: 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 “space-time 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 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, 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.

Review Copy – for publishers of book reviews
Accessibility – to request an alternate format of an AMS title
Please select which format for which you are requesting permissions.