Types of quantum field thories
In supersymmetric quantum field theory with extended supersymmetry, certain extremal supermultiplets have some of the supersymmetries retained (have 0-eigenvalue under some of the supersymmetry generators). These are called Bogomol’nyi–Prasad–Sommerfield saturated solutions.
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.
Specifically in superstring theory BPS states in target space correspond to string states on the worldsheet which are annihilated by the left-moving (say) half of the Dirac-Ramond operator. These are counted by the Witten genus, see at Witten genus – Relation to BPS states.
The BPS bound derives its name from
(E. B. Bogomolnyj) Е. Б. Богомольний, Устойчивость классических решений, Яд. Физ. 24 (1976) 449–454
M. K. Prasad, C . M. Sommerfield, Exact classical solution for ‘t Hooft monopole and the Julia-Zee dyon, Phys. Rev. Lett. 35 (1975) 760–762.
The original article identifying the role of BPS states in supersymmetric field theory is
Further developments are in
A. H. Chamseddine, M. S. Volkov, Non-abelian BPS monopoles in gauged supergravity, Physical Review Letters 79: 3343–3346 (1997) hep-th/9707176.
S. Weinberg, The quantum theory of fields, vol. II
Tudor Dimofte, Sergei Gukov, Refined, Motivic, and Quantum, arXiv:0904.1420
Davide Gaiotto, Gregory W. Moore, Andrew Neitzke, Wall-crossing, Hitchin systems, and the WKB approximation, arxiv:0907.3987
R. Pandharipande, R.P. Thomas, Stable pairs and BPS invariants, arXiv:0711.3899
Markus Reineke, Cohomology of quiver moduli, functional equations, and integrality of Donaldson-Thomas type invariants, arXiv:0903.0261
Duiliu-Emanuel Diaconescu, Moduli of ADHM sheaves and local Donaldson-Thomas theory, arXiv:0801.0820
Tom Bridgeland, Stability conditions on triangulated categories, Ann. of Math. 166 (2007) 317–345,math.AG/0212237
An introduction that starts at the beginning and then covers much of the ground in some detail is
A survey of progress on the most general picture is in