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2011•Symmetry Integrability and Geometry Methods and ApplicationsOpen access

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

Andrei V. Smilga

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Abstract

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.

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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.

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Available abstract

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.

Key concepts: Holomorphic function, Complex manifold, Hermitian manifold, Kähler manifold, Hermitian matrix, Complex conjugate, Pure mathematics, Supersymmetry

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