First of all, let me congratulate the theoretical physics community for this site. I am a mathematics student with very little background in phyiscs. The question I want to ask is:

What is the proper mathematical definition of BPS states?

In string theory the BPS states correspond either to coherent sheaves or special Lagrangians of Calabi- Yau manifold depending upon the type of string theory considered. but in SUSY quantum field theories in 4d, there are no CYs as far I know (which is very little) and in gravity theories, these corresponds to some Black Holes. So what is the general mathematical definition of BPS states which is independent of the theory in consideration, say a general SUSY Quantum field theories, be it QFT, string theory, Gravity and in any dimension.

I am very sorry, if the question is not framed properly due to my ignorance of the physics involved.
Thanks

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