### 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.

pdf (337 kb)   tex (13 kb)

References

1. Arenas R., Constructing a matrix representation of the Lie group $G_2$, Senior Thesis, Harvey Mudd College, 2005, available at https://www.math.hmc.edu/seniorthesis/archives/2005/rarenas/rarenas-2005-thesis.pdf.
2. Bryant R.L., Some remarks on $G_2$-structures, in Proceedings of Gökova Geometry-Topology Conference 2005, Gökova Geometry/Topology Conference (GGT), Gökova, 2006, 75-109, math.DG/0305124.
3. Cartan E., Nombres complexes (Exposé, d'après l'article allemand de E. Study), in Encyclopédie des sciences mathématiques pures et appliquées, Vol. I, Chap. 5, Editor J. Molk, Gauthier-Villars, 1908, 329-468.
4. Cattaneo A.S., Zambon M., A supergeometric approach to Poisson reduction, Comm. Math. Phys. 318 (2013), 675-716, arXiv:1009.0948.
5. Deligne P., Morgan J.W., Notes on supersymmetry (following Joseph Bernstein), in Quantum Fields and Strings: a Course for Mathematicians, Vols. 1, 2 (Princeton, NJ, 1996/1997), Amer. Math. Soc., Providence, RI, 1999, 41-97.
6. Harvey R., Lawson Jr. H.B., Calibrated geometries, Acta Math. 148 (1982), 47-157.
7. Marsden J., Weinstein A., Reduction of symplectic manifolds with symmetry, Rep. Math. Phys. 5 (1974), 121-130.
8. Mehta R.A., On homotopy Poisson actions and reduction of symplectic $Q$-manifolds, Differential Geom. Appl. 29 (2011), 319-328, arXiv:1009.1280.