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Operator-Algebraic Construction of Gauge Theories and Jones’ Actions of Thompson’s Groups

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Using ideas from Jones, lattice gauge theory and loop quantum gravity, we construct \(1+1\)-dimensional gauge theories on a spacetime cylinder. Given a separable compact group G, we construct localized time-zero fields on the spatial torus as a net of C*-algebras together with an action of the gauge group that is an infinite product of G over the dyadic rationals and, using a recent machinery of Jones, an action of Thompson’s group T as a replacement of the spatial diffeomorphism group. Adding a family of probability measures on the unitary dual of G we construct a state and obtain a net of von Neumann algebras carrying a state-preserving gauge group action. For abelian G, we provide a very explicit description of our algebras. For a single measure on the dual of G, we have a state-preserving action of Thompson’s group and semi-finite von Neumann algebras. For \(G={\mathbf {S}}\) the circle group together with a certain family of heat-kernel states providing the measures, we obtain hyperfinite type III factors with a normal faithful state providing a nontrivial time evolution via Tomita–Takesaki theory (KMS condition). In the latter case, we additionally have a non-singular action of the group of rotations with dyadic angles, as a subgroup of Thompson’s group T, for geometrically motivated choices of families of heat-kernel states.

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Acknowledgements

Part of this project was done when both authors were working at the University of Rome, Tor Vergata thanks to the very generous support of Roberto Longo. We are very grateful to Roberto for giving us this great opportunity besides his constant encouragement and support during various stages of the project. AS thanks the Alexander-von-Humboldt Foundation for generous financial support during his stay at the University of Rome, Tor Vergata. Moreover, AS acknowledges financial support and kind hospitality by the Isaac Newton Institute and the Banff International Research Station where parts of this work were developed. Furthermore, we are grateful to Thomas Thiemann, Yoh Tanimoto, Luca Giorgetti and Vincenzo Morinelli for comments and discussions during various stages of this work. Finally, we are grateful to the referee for constructive comments which improved the clarity of this manuscript.

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Authors and Affiliations

  1. School of Mathematics and Statistics, University of New South Wales, Sydney, NSW, 2052, Australia

    Arnaud Brothier

  2. Department of Mathematics, University of Rome Tor Vergata, Via della Ricerca Scientifica, 00133, Rome, Italy

    Alexander Stottmeister

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  1. Arnaud Brothier
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Correspondence to Arnaud Brothier.

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Communicated by Y. Kawahigashi

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Both authors were partially supported by European Research Council Advanced Grant 669240 QUEST. AS was supported by Alexander-von-Humboldt Foundation through a Feodor Lynen Research Fellowship. AB is supported by a University of New South Wales Sydney starting Grant.

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Brothier, A., Stottmeister, A. Operator-Algebraic Construction of Gauge Theories and Jones’ Actions of Thompson’s Groups. Commun. Math. Phys. 376, 841–891 (2020). https://doi.org/10.1007/s00220-019-03603-4

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