Harmonic $N = 2$ superspace was discovered in Dubna in 1984 [1,2] as the powerful unique tool of the geometric superfield off-shell description of $N = 2$, 4D supersymmetric field theories with the maximal spins 1, 2, and 1/2 ($N = 2$ Yang-Mills theories, supergravity and matter hypermultiplets). In my talk I give a brief account of the basic achievements of the harmonic methods (not only in supersymmetry), including the newest applications in $N = 2$ theories of higher spins with $s > 2$. The harmonic superspace approach makes it possible to set up the previously unknown off-shell formulations of the gauge $N = 2$ higher-spin theories in terms of the unconstrained analytic $N = 2$ superfields, in both conformal and non-conformal cases [3-5]. It is expounded how to deduce N = 2 higher-spin theories on the curved AdS4 background, proceeding from the harmonic description of the superconformal theories [6]. The basic idea consists in introducing a constant isotriplet cik that breaks $N = 2$ superconformal symmetry $SU(2, 2|2)$ down to the AdS4 supersymmetry $OSp(2|4;R)$.
References
[1] A. S. Galperin, E. A. Ivanov, S. Kalitzin, V. I. Ogievetsky, E. S. Sokatchev, Unconstrained $N = 2$ Matter, Yang-Mills and Supergravity Theories in Harmonic Superspace, Class. Quant. Grav. 1 (1984) 469-498;
[2] A. S. Galperin, E. A. Ivanov, V. I. Ogievetsky, E. S. Sokatchev, Harmonic superspace, Cambridge Monographs on Mathematical Physics, Cambridge University Press, 2001, 306 p.;
[3] I. Buchbinder, E. Ivanov, N. Zaigraev, Unconstrained off-shell superfield formulation of 4D, N = 2 supersymmetric higher spins, JHEP 12 (2021) 016 [arXiv:2109.07639 [hep-th]];
[4] E. Ivanov, Higher Spins in Harmonic Superspace, Teor. Mat. Fiz. 217 (2023) 515-532 [Theor. Math. Phys. 217 (2023) 1855-1869], [arXiv:2306.10401 [hep-th]];
[5] I. Buchbinder, E. Ivanov, N. Zaigraev, N = 2 superconformal higher-spin multiplets and their hypermultiplet couplings, JHEP 08 (2024) 120 [arXiv:2404.19016 [hep-th]];
[6] E. Ivanov, N. Zaigraev, 2025, in preparation.