In the classical bosonic sector of a supersymmetric field theory, the Bogomol'nyi–Prasad–Sommerfield (BPS) bound (named after Evgeny Bogomolny, M.K.
Prasad, and Charles Sommerfield[1][2]) provides a lower limit on the energy of static field configurations, depending on their topological charges or boundary conditions at spatial infinity.
This bound manifests as a series of inequalities for solutions of the classical bosonic field equations.
Their existence and properties are deeply connected to the underlying supersymmetry of the theory, even though the bound itself can be formulated within the bosonic sector alone.
The acronym BPS stands for Bogomol'nyi, Prasad, and Sommerfield, who first derived the bound in the context of magnetic monopoles in Yang-Mills theory in 1975.
Supersymmetry is a theoretical framework that relates bosons and fermions, particles with integer and half-integer spin, respectively.
The supersymmetry algebra includes anticommutators of these generators, which typically involve the Hamiltonian (energy operator)
[3][4] The BPS bound arises from the positivity of the norm of states in the Hilbert space.
They provide a window into the non-perturbative regime of these theories and allow for exact calculations of certain quantities.
BPS states have played a crucial role in the development of string theory, the AdS/CFT correspondence, and our understanding of black hole physics.
The concept of the BPS bound first arose in the study of magnetic monopoles in non-abelian gauge theories.
Specifically, it was shown that the mass of a 't Hooft-Polyakov monopole and a Julia-Zee dyon is bounded from below by a quantity proportional to its topological charge.
Solutions that saturate this bound are called BPS monopoles or BPS dyons, and they possess special properties and play a significant role in both classical and quantum field theory.
The 't Hooft-Polyakov monopole is a static, finite-energy solution in a non-abelian gauge theory, typically SU(2), spontaneously broken to U(1) by a scalar Higgs field in the adjoint representation.
Solutions to these first-order equations are the BPS monopoles, which have the minimal energy for a given magnetic charge.
Solutions to these equations are the BPS dyons, which have the minimal energy for given electric and magnetic charges.
The BPS monopoles and dyons are important because they are stable, finite-energy solutions that saturate a classical energy bound.
Their existence and properties are closely related to the topology of the gauge group and the Higgs field.
In supersymmetric extensions of the theory, BPS states correspond to short representations of the supersymmetry algebra and are protected from quantum corrections.
The Reissner-Nordström metric describes the spacetime geometry around a spherically symmetric, electrically charged black hole.
The ADM mass, a concept from general relativity that defines the total mass-energy of an asymptotically flat spacetime, is simply
The event horizon(s) of the Reissner-Nordström black hole are located at the radial coordinates where the metric component
In the context of supergravity theories (supersymmetric extensions of general relativity), extremal Reissner-Nordström black holes can be interpreted as BPS states.
can be viewed as a saturation of a classical inequality, analogous to the BPS bound in supersymmetric field theories.
In general relativity, various energy conditions are imposed on the stress-energy tensor
For the electromagnetic field, which is the source of the Reissner-Nordström black hole's charge, the weak energy condition translates to:
supergravity, the extremal Reissner-Nordström black hole is not just an analog but a true BPS state.
This means that some of the supersymmetry transformations leave the black hole solution invariant.
The fact that extremal black holes can be BPS states has profound implications.
It suggests a deep connection between gravity, supersymmetry, and the quantum nature of spacetime.