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

TL;DR

This paper demonstrates that the Dolbeault complex can be defined on S^4 minus a point and S^6 using supersymmetry, calculating the spectrum and index, revealing new geometric insights.

Contribution

It introduces a supersymmetric approach to define the Dolbeault complex on non-complex manifolds like S^4 minus a point and S^6, and computes their spectra and indices.

Findings

Dolbeault complex defined on S^4 minus a point and S^6 using supersymmetry

Spectrum of the Dolbeault Laplacian computed, showing 3 bosonic zero modes on S^4 minus a point

Dolbeault index on S^4 minus a point is equal to 3

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.