Casimir operators of $4D,\mathcal{N}=2$ supersymmetry in the harmonic approach

Abstract

We construct Casimir operators of $4D,\mathcal{N}=2$ supersymmetry algebra in the harmonic superspace approach, and provide the explicit expressions for them in terms of covariant derivatives in the analytic basis. Specifically, the superisospin operator $C_I$ is expressed in terms of new 'long' harmonic derivatives ${\nabla }^{\pm \pm },{\nabla }^0$ as $C_I=\frac{1}{4}{\left({\nabla }^0\right)}^2+\frac{1}{2}\left\{{\nabla }^{++},{\nabla }^{--}\right\}$. These derivatives involve non-local terms with the inverse Box operator ${\square }^{-1}$ and satisfy modified $SU(2)$ algebra relations in which the eigenvalues of ${\nabla }^0$ on analytic superfields are shifted by -2 relative to the standard $U(1)$ harmonic charges. We also construct $\mathcal{N}=2$ superhelicity and super-isohelicity operators relevant to the massless case (with vanishing $\square$) and find their eigenvalues for few instructive examples of $4D,\mathcal{N}=2$ superfield theories.

arXiv comment0 + 38 pages, typos corrected