Finite higher spin transformations from exponentiation

Abstract

We study the exponentiation of elements of the gauge Lie algebras $\mathrm{hs}(\lambda )$ of three-dimensional higher spin theories. Exponentiable elements generate one-parameter groups of finite higher spin symmetries. We show that elements of $\mathrm{hs}(\lambda )$ in a dense set are exponentiable, when pictured in certain representations of $\mathrm{hs}(\lambda )$, induced from representations of $SL(2,\mathbb{R})$ 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 $HS(\lambda )$ of finite higher spin symmetries. Interestingly, this construction is possible only for $0\le \lambda \le 1$, 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 $\mathrm{hs}(\lambda )$. Among those, we identify families of elements of $\mathrm{hs}(\lambda )$ exponentiating to the unit of $HS(\lambda )$, 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.

34 pages. v3: references added. Added a discussion of the Euclidean higher spin symmetry group. Unlike what was claimed in a previous version, the formalism developed here can be applied to the Euclidean case as well

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• Black Holes in the 3D Higher Spin Theory and Their Quasi Normal Modes
• Higher-spin gauge theories in three spacetime dimensions
• On matter coupled to the higher spin square