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Finite Higher Spin Transformations from Exponentiation

Monnier, Samuel

In: Communications in Mathematical Physics, 2015, vol. 336, no. 1, p. 1-26

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    Title
    • Finite Higher Spin Transformations from Exponentiation
    Author
    • Monnier, Samuel. Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190, 8057, Zurich, Switzerland
    Document Type
    • Postprint
    Language
    • English
    Published in
    • Communications in Mathematical Physics, 2015, vol. 336, no. 1, p. 1-26. Springer Berlin Heidelberg
    Other Version
    Classification
    • Mathematics
    OAI-PMH Identifier
    • oai:doc.rero.ch:332241
    Summary
    We study the exponentiation of elements of the gauge Lie algebras hs(λ) of three-dimensional higher spin theories. Exponentiable elements generate one-parameter groups of finite higher spin symmetries. We show that elements of hs(λ) in a dense set are exponentiable, when pictured in certain representations of hs(λ), induced from representations of $${SL(2,\mathbb{R})}$$ S L ( 2 , R ) in the complementary series. We also provide a geometric picture of higher spin gauge transformations clarifying the physical origin of these representations. This allows us to construct an infinite-dimensional topological group HS(λ) of finite higher spin symmetries. Interestingly, this construction is possible only for 0 ≤λ ≤ 1, which are the values for which the higher spin theory is believed to be unitary and for which the Gaberdiel-Gopakumar duality holds. We exponentiate explicitly various commutative subalgebras of hs(λ). Among those, we identify families of elements of hs(λ) exponentiating to the unit of HS(λ), generalizing the logarithms of the holonomies of BTZ black hole connections. Our techniques are generalizable to the Lie algebras relevant to higher spin theories in dimensions above three.