We revisit the issue of higher-dimensional counterterms for the \( \mathcal{N}=\left(1,1\right) \) supersymmetric Yang-Mills (SYM) theory in six dimensions using the off-shell \( \mathcal{N}=\left(1,0\right) \) and on-shell \( \mathcal{N}=\left(1,1\right) \) harmonic superspace approaches. The second approach is developed in full generality and used to solve, for the first time, the \( \mathcal{N}=\left(1,1\right) \) SYM constraints in terms of \( \mathcal{N}=\left(1,0\right) \) superfields. This provides a convenient tool to write explicit expressions for the candidate counterterms and other \( \mathcal{N}=\left(1,1\right) \) invariants and may be conducive to proving non-renormalization theorems needed to explain the absence of certain logarithmic divergences in higher-loop contributions to scattering amplitudes in \( \mathcal{N}=\left(1,1\right) \) SYM.

We show that the recently constructed higher-derivative 6D SYM theory involves internal chiral anomaly breaking gauge invariance. The anomaly is cancelled when adding to the theory an adjoint matter hypermultiplet.

We present a Lagrangian formulation for \( \mathcal{N} = 4 \) supersymmetric quantummechanical systems describing the motion in external non-Abelian self-dual gauge fields. For any such system, one can write a component supersymmetric Lagrangian by introducing extra bosonic variables with topological Chern-Simons type interaction. For a special class of such system when the fields are expressed in the ‘t Hooft ansatz form, it is possible to give a superfield description using harmonic superspace formalism. As a new explicit example, the \( \mathcal{N} = 4 \) mechanics with Yang monopole is constructed.

We explore “weak” supersymmetric systems whose algebra involves, besides Poincare generators, extra bosonic generators not commuting with supercharges. This allows one to have inequal number of bosonic and fermionic 1-particle states in the spectrum. Coleman–Mandula and Haag–Lopuszanski–Sohnius theorems forbid the presence of such extra bosonic charges in interacting theory for d⩾3. However, these theorems do not apply in one or two dimensions. For d=1, we construct a nontrivial interacting system characterized by weak supersymmetric algebra. It is related to “n-fold” supersymmetric systems and to quasi-exactly solvable systems studied earlier.

We present a pedagogical review of our current understanding of the ultraviolet structure of N = (1,1) 6D supersymmetric Yang–Mills theory and of N = 8 4D supergravity. These theories are not renormalizable, they involve power ultraviolet divergences and, in all probability, an infinite set of higherdimensional counterterms that contribute to on-mass-shell scattering amplitudes. A specific feature of supersymmetric theories (especially, of extended supersymmetric theories) is that these counterterms may not be invariant off shell under the full set of supersymmetry transformations. The lowest-dimensional nontrivial counterterm is supersymmetric on shell. Still higher counterterms may lose even the on-shell invariance. On the other hand, the full effective Lagrangian, generating the amplitudes and representing an infinite sum of counterterms, still enjoys the complete symmetry of original theory. We also discuss simple supersymmetric quantum-mechanical models that exhibit the same behaviour.