Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 12 (2016), 116, 6 pages      arXiv:1603.06603      https://doi.org/10.3842/SIGMA.2016.116
Contribution to the Special Issue “Gone Fishing”

The Quaternions and Bott Periodicity Are Quantum Hamiltonian Reductions

Theo Johnson-Freyd
Perimeter Institute for Theoretical Physics, Waterloo, ON, Canada

Received August 30, 2016, in final form December 09, 2016; Published online December 11, 2016

Abstract
We show that the Morita equivalences $\mathrm{Cliff}(4) \simeq {\mathbb H}$, $\mathrm{Cliff}(7) \simeq \mathrm{Cliff}(-1)$, and $\mathrm{Cliff}(8) \simeq {\mathbb R}$ arise from quantizing the Hamiltonian reductions ${\mathbb R}^{0|4} // \mathrm{Spin}(3)$, ${\mathbb R}^{0|7} // G_2$, and ${\mathbb R}^{0|8} // \mathrm{Spin}(7)$, respectively.

Key words: Clifford algebras; quaternions; Bott periodicity; Morita equivalence; quantum Hamiltonian reduction; super symplectic geometry.

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