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Invariant Operators on Supermanifolds and Standard Models

Pavel Grozman
Organization: University of Stockholm
Department: Mathematics
Dimitry Leites
Organization: University of Stockholm
Department: Mathematics
Irina Shchepochkina
Organization: The Independent University of Moscow
Department: Mathematics
Journal / Anthology

In book, "Multiple facets of quantization and supersymmetry" World Scientific Publishing, River Edge, NJ
Year: 2002
Page range: 508-555

Here we continue to list the differential operators invariant with respect to the 15 exceptional simple Lie superalgebras $\fg$ of polynomial vector fields. A part of the list (for operators acting on tensors with finite dimensional fibers) was earlier obtained in 2 of the 15 cases by Kochetkov and in one more instance by Kac and Rudakov. Broadhurst conjectured that some of these structures pertain to The Standard Models of elementary particles. So, the Grand Unified Theory, if exists, will be formulated in terms of operators we found, or their $r$-nary analogs. Calculations are performed with the aid of Grozman's {\it Mathematica}-based SuperLie package. When degeneracy conditions are violated (absence of singular vectors) the corresponding module of tensor fields is irreducible. We also verified some of the earlier findings.

*Mathematics > Algebra > Group Theory

representations of Lie superalgebras, exceptional Lie superalgebras, invariant operators