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Invariant operators on supermanifolds and Standard Models

Pavel Grozman, Dimitry Leites, Irina Shchepochkina

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.

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.

Paper Structure

This paper contains 23 sections, 179 equations.

Table of Contents

  1. To the memory of Misha Marinov (by D.L.)
  2. Introduction
  3. Veblen's problem
  4. Rudakov's breakthrough (following Bernstein$^{5}$)
  5. Further ramifications of Veblen's problem
  6. Arity $>1$
  7. Generalized tensors and primitive forms
  8. Superization leads to new developments
  9. An infinitesimal version of Veblen's problem
  10. Operators invariant with respect to nonstandard realizations
  11. On description of irreducible ${\cal L}$-modules
  12. Brief description of the exceptional algebras
  13. Singular vectors for ${\mathfrak{g}}={\mathfrak{v}}{\mathfrak{le}}(3|6)$
  14. Singular vectors for ${\mathfrak{g}}={\mathfrak{v}}{\mathfrak{le}}(4|3)$
  15. Singular vectors for ${\mathfrak{g}}={\mathfrak{m}}{\mathfrak{b}}(4|5)$ (after Kochetkov)
  16. ...and 8 more sections