Local boundary conditions involving field strengths and the normal to the boundary, originally studied in anti-de Sitter space-time, have been recently considered in one-loop quantum cosmology. This paper derives the conditions under which spin-lowering and spin-raising operators preserve these local boundary conditions on a 3-sphere for fields of spin 0, 1 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203734889/cc82ad37-66a9-4d5e-91f1-ce4dcbddbaf7/content/inequ140.tif"/> , 1, 3 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203734889/cc82ad37-66a9-4d5e-91f1-ce4dcbddbaf7/content/inequ141.tif"/> and 2. Moreover, the two-component spinor analysis of the four potentials of the totally symmetric and independent field strengths for spin 3 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203734889/cc82ad37-66a9-4d5e-91f1-ce4dcbddbaf7/content/inequ142.tif"/> is applied to the case of a 3-sphere boundary. It is shown that such boundary conditions can only be imposed in a fiat Euclidean background, for which the gauge freedom in the choice of the potentials remains. Alternative boundary conditions for supergravity involving the spinor-valued 1-forms for gravitinos and the normal to the boundary are also studied.