Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 7 (2011), 105, 14 pages      arXiv:1105.3935      https://doi.org/10.3842/SIGMA.2011.105

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

Andrei V. Smilga
SUBATECH, Université de Nantes, 4 rue Alfred Kastler, BP 20722, Nantes 44307, France

Received June 22, 2011, in final form November 09, 2011; Published online November 15, 2011

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.

Key words: Dolbeault; supersymmetry.

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