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Dolbeault Complex on $S^4\setminus \{\,\cdot\,\}$ and $S^6\setminus\{\,\cdot\,\}$ through Supersymmetric Glasses
Symmetry, integrability and geometry: methods and applications, Tome 7 (2011) Cet article a éte moissonné depuis la source Math-Net.Ru

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$S^4$ 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 $\partial$ 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 $S^4\setminus\{\,\cdot\,\}$ is equal to $3$.
Keywords: supersymmetry.
Mots-clés : Dolbeault 
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     author = {Andrei V. Smilga},
     title = {Dolbeault {Complex} on $S^4\setminus \{\,\cdot\,\}$ and $S^6\setminus\{\,\cdot\,\}$ through {Supersymmetric} {Glasses}},
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     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2011_7_a104/}
}
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Andrei V. Smilga. Dolbeault Complex on $S^4\setminus \{\,\cdot\,\}$ and $S^6\setminus\{\,\cdot\,\}$ through Supersymmetric Glasses. Symmetry, integrability and geometry: methods and applications, Tome 7 (2011). http://geodesic.mathdoc.fr/item/SIGMA_2011_7_a104/

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