What is the mathematical definition of Bogomol'nyi–Prasad–Sommerfield (BPS) states, independent of any specific physical theory.


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It's a state annihilated by $E-Q$ where $E$ is the energy, or another dynamical/isometry generator, and $Q$ is the sum of multiples of other conserved charges such that one may write $$ E - Q = \sum_{i,j} c_{ij}\{Q_i,Q_j\} $$ i.e. the difference between energy and charges may be obtained as an anticommutator of some supercharges – Grassmann-odd symmetry generators acting on the system. It's easy to show, by looking at the vanishing expectation value of $E-Q$, that BPS states are equivalently those that are annihilated by some supercharges $Q_i$ or their combinations. Consequently, the space of states obtained by the action of all the generators on the BPS state has a smaller dimension than the dimension of the long multiplet – $2^{N_{\rm sup}/2}$. We call such multiplets short.


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