BPS state (Rev #2) in nLab

In supergravity theory, certain extremal solutions have some of the supersymmetries retained. These are the so-called Bogomol’nyi–Prasad–Sommerfield saturated solutions which exist also in some models of soliton theory (English Wikipedia: Bogomol’nyi–Prasad–Sommerfield bound). Namely, the fact that a certain fraction (typically one half or fourth of supersymmetry generators) of supersymmetry is retained implies the saturation of the BPS-bound, which does make sense a bit more generally. The retained generators generate a nontrivial subalgebra of the full supersymmetry algebra and carry conserved charges; the mass is exactly determined in terms of these charges.

In geometric models, like variants of the superstring theory, it is very important to investigate moduli spaces of classical vacua (e.g. the ground states for the D-brane systems). BPS-states correspond just to a part of the moduli problem which is often the most tractable.

Several mathematical theories in geometry are interpreted as counting BPS-states in the sense of integration on appropriate compactification of the moduli space of BPS-states in a related physical model attached to the underlying geometry: most notably the Gromov–Witten invariants, Donaldson–Thomas invariants and the Thomas–Pandharipande invariants; all the three seem to be deeply interrelated though they are defined in rather very different terms. The compactification of the moduli space involves various stability conditions.

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