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Abelian 6d (2,0) theory has SO(5) R symmetry. We twist this theory byidentifying the R symmetry group with the SO(5) subgroup of the SO(1,5) Lorentzgroup. This twisted theory can be put on any five-manifold M, times R, whilepreserving one scalar supercharge. We subsequently assume the existence of oneunit normalized Killing vector field on M, and we find a corresponding SO(4)twist that preserves two supercharges and is a generalization of the geometricLanglands twist of 4d SYM. We generalize the story to non-Abelian gauge groupfor the corresponding 5d SYM theories on M. We derive a vanishing theorem forBPS contact instantons by identifying the 6d potential energy and its BPSbound, in the 5d theory. To this end we need to perform a Wick rotation thatcomplexifies the gauge field.

The geometric Langlands twist in five and six dimensions