Invariant operators on supermanifolds and Standard Models

Abstract

Here we continue to list the differential operators invariant with respect to the 15 exceptional simple Lie superalgebras 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 and Kac conjectured that some of these structures pertain to the Standard Models of elementary particles and the Grand Unified Theories. So, GUT, if any exists, will be formulated in terms of operators we found, or their r-nary analogs to be found. Calculations are performed with the aid of Grozman's Mathematica-based SuperLie package. When degeneracy conditions are violated (absence of singular vectors) the corresponding module of tensor fields is irreducible. We consider generalized tensors corresponding to fibers of countable infinite dimension, not only finite dimensional ones. We also verified some of the earlier findings.

48 p., Latex; typos in Abstract and address are corrected. In: M. Olshanetsky, A. Vainstein (eds.) {\em Multiple facets of quantization and supersymmetry Michael Marinov Memorial Volume}, World Sci., 2002 to appear

References in corpus (1)

• Indecomposable representations of Lie superalgebras

Cited by in corpus (18)

• Differential operators on supercircle: conformally equivariant quantization and symbol calculus
• Cohomology of the Lie Superalgebra of Contact Vector Fields on ${\mathbb{R}}^{1|1}$ and Deformations of the Superspace of Symbols
• Towards classification of simple finite dimensional modular Lie superalgebras
• Structures of G(2) type and nonintegrable distributions in characteristic p
• Global Analytic Approach to Super Teichmueller Spaces
• Second-Order Conformally Equivariant Quantization in Dimension 1|2
• Projectively equivariant quantizations over the superspace ${\mathbb{R}}^{p|q}$
• The space of m-ary differential operators as a module over the Lie algebra of vector fields
• Cartan matrices and presentations of Cunha and Elduque superalgebras
• On osp(p+1,q+1|2r)-equivariant quantizations
• Symmetries of modules of differential operators
• The ternary invariant differential operators acting on the spaces of weighted densities
• Exceptional Lie Superalgebras, Invariant Morphisms, and a Second-Gauged Standard Model
• Divided power (co)homology. Presentations of simple finite dimensional modular Lie superalgebras with Cartan matrix
• New simple modular Lie superalgebras as generalized prolongs
• Lie antialgebras: cohomology and representations
• Riemannian Curl in Contact Geometry
• Maximal depth implies su(3)+su(2)+u(1)