## 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.

### MSC:

 81T60 Supersymmetric field theories in quantum mechanics 83E50 Supergravity 11R52 Quaternion and other division algebras: arithmetic, zeta functions 17A35 Nonassociative division algebras 17A70 Superalgebras

### Citations:

Zbl 1210.81117; Zbl 0984.00503
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