Dolbeault Complex on S⁴\{·} and S⁶\{·} through Supersymmetric Glasses

Abstract

S⁴ is not a complex manifold, but it is sufficient to remove one point to make it complex. Using supersymmetry methods, we show that the Dolbeault complex (involving the holomorphic exterior derivative ∂ and its Hermitian conjugate) can be perfectly well defined in this case. We calculate the spectrum of the Dolbeault Laplacian. It involves 3 bosonic zero modes such that the Dolbeault index on S⁴\{·} is equal to 3.