Bogomol'nyi-Prasad-Sommerfield (BPS) states: Mathematical definition 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.
 A: In any supersymmetric theory, a BPS state is a state which preserves some of the supersymmetry.
If we take as a definition of a supersymmetric theory, some theory (classical or quantum) which admits a Lie superalgebra of symmetries, then a BPS state (or configuration) of such a theory is one which is annihilated by some nonzero odd element in the superalgebra.
Of course, the original meaning comes from the study of magnetic monopoles.  Solutions of the Bogomol'nyi equation are precisely those which saturate a bound, the so-called Bogomol'nyi-Prasad-Sommerfield (BPS) bound.
The relation between the two notions is related to the fact that monopole configurations can be thought of as configurations in a four-dimensional $N=2$ supersymmetric gauge theory which preserve half of the supersymmetry.
A: The BPS bound was discovered independently of supersymmetry, but it was then better understood as general feature of the supersymmetry algebra. Look at the original paper by Witten and Olive. BPS states are states which saturate the BPS bound, forming "short" representations an extended supersymmetry algebra. Such representations have special properties, which often can be thought of as consequence of some fraction of the supersymmetry which remains in their presence. The examples you cite are special cases, in all those cases the special property of the objects you mention is a deduced from the requirement that the states involved form a short representation of the supersymmetry algebra.
