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**Division algebras and supersymmetry. II.**
*(English)*
Zbl 1260.81222

Deep connections between supersymmetry and the four normed division algebras
(real numbers, complex numbers, quaternions and octonions) have been studied
in [J. C. Baez and J. Huerta, Proceedings of Symposia in Pure Mathematics 81, 65–80 (2010; Zbl 1210.81117)], where it was shown how an \(n\)-dimensional normed division
algebra gives rise to vectors and spinors in \((n+2)\)-dimensional spacetime,
thereby yielding the 3-\(\psi\)’s rule. In this paper the authors construct
vectors and spinors in \((n+3)\)-dimensional spacetime, thereby yielding the
4-\(\Psi\)’s rule. They show also that the 3-\(\psi\)’s rule and 4-\(\Psi\)’s rule
can be interpreted as cocycle conditions. In every dimension, a symmetric
bilinear intertwining operator eating two spinors and then splitting out a
vector gives rise to a super-Minkowski spacetime. The infinitesimal
translation symmetries of this entity constitute a Lie superalgebra called
supertranslation algebra in [Quantum fields and strings: a course for mathematicians. Vol. 1, 2. Material from the Special Year on Quantum Field Theory held at the Institute for Advanced Study, Princeton, NJ, 1996–1997. Providence, RI: AMS, American Mathematical Society (1999; Zbl 0984.00503)], whose third and fourth
cohomologies are respectively nontrivial in dimensions 3, 4, 6 and 10 in deep
deference to the 3-\(\psi\)’s rule, and in dimensions 4, 5, 7 and 11 in deep
deference to the 4-\(\Psi\)’s rule. It is shown that an \((n+1)\)-cocycle on a Lie superalgebra naturally gives rise to a Lie \(n\)-superalgebra
as an extension, thereby obtaining Lie \(2\)-superalgebras and Lie
\(3\)-superalgebras as extensions of supertranslation algebras. The final
section is devoted to constructing superstring Lie \(2\)-algebras and the
\(2\)-brane Lie \(3\)-algebras.

Reviewer: Hirokazu Nishimura (Tsukuba)