The σ− Cohomology Analysis for Symmetric Higher-Spin Fields
Abstract
:1. Introduction
2. Fronsdal Theory
2.1. Metric Formulation
2.2. Frame-Like Formulation
2.2.1. Tensor Formalism
2.2.2. Spinor Language in
3. The Idea of Cohomology Analysis: Example of Integer Spin Massless Fields
- (1)
- Differential gauge symmetry parameters ε span
- (2)
- Nontrivial dynamical fields span
- (3)
- Physically distinguishable differential field equations on the nontrivial dynamical fields, contained in , span
4. A Method for Calculating Cohomology
- . The harmonic cocycles annihilated by are those and only those, that are d-closed and ∂-closed simultaneously;
- . In other words, for any vector there exists a unique Hodge decomposition , where and are some vectors in V, and h is harmonic .
5. Cohomology in Minkowski Space of Any Dimension
5.1. Generating Functions
5.2. Example
5.2.1.
5.2.2. ,
5.3. Case
5.3.1. Irreducibility Conditions
5.3.2.
5.3.3.
5.3.4. ,
5.3.5. Summary
6. Cohomology in in the Spinor Language
7. Bosonic Case in
7.1.
7.2.
7.2.1. in the Diagonal Sector
7.2.2. in the Far-from-Diagonal Sector
7.2.3. Subtlety in the Near-Diagonal Sector
7.2.4. in the Near-Diagonal Sector
7.3.
7.3.1. in the Far-From-Diagonal Sector
7.3.2. on the Diagonal
7.3.3. in the Near-Diagonal Sector
7.4. Summary for Bosonic
8. Fermionic HS Fields in
8.1. Fermionic
8.2. Fermionic
8.3. Fermionic
9. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
Appendix A
Appendix A.1. Index Conventions
Appendix A.2. Coefficients in the Tensor form of the Diagram (n − 1,m − 1;p − 2)
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Bychkov, A.S.; Ushakov, K.A.; Vasiliev, M.A. The σ− Cohomology Analysis for Symmetric Higher-Spin Fields. Symmetry 2021, 13, 1498. https://doi.org/10.3390/sym13081498
Bychkov AS, Ushakov KA, Vasiliev MA. The σ− Cohomology Analysis for Symmetric Higher-Spin Fields. Symmetry. 2021; 13(8):1498. https://doi.org/10.3390/sym13081498
Chicago/Turabian StyleBychkov, Alexey S., Kirill A. Ushakov, and Mikhail A. Vasiliev. 2021. "The σ− Cohomology Analysis for Symmetric Higher-Spin Fields" Symmetry 13, no. 8: 1498. https://doi.org/10.3390/sym13081498
APA StyleBychkov, A. S., Ushakov, K. A., & Vasiliev, M. A. (2021). The σ− Cohomology Analysis for Symmetric Higher-Spin Fields. Symmetry, 13(8), 1498. https://doi.org/10.3390/sym13081498