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For a Kähler manifold, the graded algebra generated by $\partial,\overline{\partial},\partial^*,\overline{\partial}^\ast$, the Lefschetz operator, and the dual Lefschetz operator, is called the supersymmetric algebra, or in physics parlance the $N=(2,2)$ SUSY algebra. Am I correct in understanding that one gets the same algebra for all Kähler manifolds?

This post imported from StackExchange MathOverflow at 2014-08-10 19:58 (UCT), posted by SE-user Jean Delinez

As a physicist I know the following.

My sub-question is how and why this physicist's proof is mathematically imprecise. In particular, what are the gaps that need to be filled in.

@RyanThorngren @40227 Any comments to my sub-question?

What statement are you interpreting this argument as a proof of?

Mostly statement 1 established the connection between supersymmetry and Kahlerity of the target space manifold.

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