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On Universal Eigenvalues of the Casimir Operator

  • PHYSICS OF ELEMENTARY PARTICLES AND ATOMIC NUCLEI. THEORY
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Abstract

Motivated by the universal knot polynomials in the gauge Chern–Simons theory, we show that the values of the second Casimir operator on an arbitrary power of Cartan product of \({{X}_{2}}\) and the adjoint representations of simple Lie algebras can be represented in a universal form. We show that it complies with the Perelomov–Popov formula for the generating function for the Casimir spectra. We discuss the phenomenon of non-zero universal values of the Casimir operator on zero representations.

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Notes

  1. Note, that for \({{G}_{2}}\) algebra the \({{\omega }_{2}}\) weight corresponds to the long root.

  2. In [18] the Killing form is defined as \({\text{Tr}}({{\hat {X}}^{a}},{{\hat {X}}^{b}})\) in the fundamental representation, while our normalization (so called Cartan–Killing normalization) corresponds to the Killing form, defined as \({\text{Tr}}(ad{{\hat {X}}^{a}},ad{{\hat {X}}^{b}})\), i.e. in the adjoint representation.

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ACKNOWLEDGMENTS

I am grateful to Professor R. Mkrtchyan for his valuable guidance and productive discussions. This work was fulfilled within the Regional Doctoral Program on Theoretical and Experimental Particle Physics sponsored by VolkswagenStiftung and was partially supported by the Science Committee of the Ministry of Science and Education of the Republic of Armenia under contract 18T-1C229.

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  1. Yerevan Physics Institute, Yerevan, Armenia

    M. Y. Avetisyan

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  1. M. Y. Avetisyan
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Avetisyan, M.Y. On Universal Eigenvalues of the Casimir Operator. Phys. Part. Nuclei Lett. 17, 779–783 (2020). https://doi.org/10.1134/S1547477120050039

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