Dolbeault Complex on S^4{·} and S^6{·} through Supersymmetric Glasses

Abstract

S^4 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 S^4{·} is equal to 3.

Keywords

• Dolbeault
• supersymmetry