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.
More in detail, where in a plain supersymmetry super Lie algebra a suitable basis of supersymmetry generators has odd bracket proportional to the spacetime translation and hence to an energy/mass operator (with terminology as at unitary representation of the Poincaré group)
for extended supersymmetry there are further bosonic super Lie algebra generators (charges) such that
If follows from the supersymmetry algebra that is a positive definte bilinear form, which puts a lower bound on the energy given the values of these extra charges. This is called the BPS bound.
In partciular when this bound is satisfied in that some of the eigenvalues of the matrix are actually equal to the energy/mass, then the corresponding component of the right hand side in the above equation vanishes and hence then the corresponding supersymmetry generators may annihilate the given state, then called a BPS state. This way enhanced supersymmetry of states goes along with certain charges taken extremal values.
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 following are some observations on the formalization of BPS states from the nPOV, in higher differential geometry. More details are in classicalinhigher, section 3.3 and dcct, section 188.8.131.52.3. See also current algebra – As a homotopy Lie algebra.
Consider then a super-spacetime locally modeled on as a Cartan geometry, solving the relevant supergravity equations of motion (e.g. 11-dimensional supergravity for , heterotic supergravity for and , type IIA supergravity for and or type IIB supergravity for ).
This means in particular that carries a super differential form
By (AGIT 89) is a BPS state to the extent that it carries Killing spinors which form a central Lie algebra extension of a sub-algebra of the supersymmetry algebra (i.e. of the super translation Lie algebra) by which is classified by the cocycle given by
Now we observe that by (hgpII, theorem 3.3.1) this is precisely the 0-truncation of the super-Poisson bracket Lie n-algebra induced by regarding as an pre-n-plectic supermanifold and restricting along the inclusion of the Killing vectors/Killing spinors into all the Hamiltonian vector fields.
The elements in here are precisely the -brane charges, as discussed in (AGIT 89, p. 8).
Hence is the more BPS the more odd-graded elements there are in (or its restriction to super-isometries). Hence is a 1/2 BPS state of supergravity if the odd dimension of this is half that of , it is 1/4 BPS if the odd dimension is one fourth of that of , etc.
Notice that if
is a prequantization of , i.e. an actual WZW term with curvature , then is supposed to be the Lie differentiation of the stabilizer group of , which is the quantomorphism n-group . (This Lie differentiation statement is strictly shown only for and in dcct but clearly should hold generally.)
Hence we may regard (or its restriction to super-isometries) as the Lie integration of the brane-charge extended supersymmetry algebra. By the discussion at conserved current – In higher differential geometry this is indeed the n-group of conserved currents of regraded as a local Lagrangian, and so this conceptually connects back to the considerations in (AGIT 89).
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, Charles 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
Steven Weinberg, The quantum theory of fields, vol. II
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
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
Andrew Callister, Topological BPS charges in 10- and 11-dimensional supergravity, thesis 2010 (spire)
Cristine N. Ferreira, BPS solution for eleven-dimensional supergravity with a conical defect configuration (arXiv:1312.0578)
The conceptual identification of the relevant brane-charge extension of the supersymmetry algebra as that of the conserved currents of the Green-Schwarz super p-brane sigma models for branes is due to
The generalization oft this perspective to the M5-brane is discussed in
and the generalizatin to D-branes is discussed in
The structure found in that article matches the general extension structure discussed in
in view of
Discussion of the relation of that to the traditional discussion of current algebras is in