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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 $\{Q_A\}$ of supersymmetry generators has odd bracket proportional to the spacetime translation and hence to an energy/mass operator $E$ (with terminology as at unitary representation of the Poincaré group)
for extended supersymmetry there are further bosonic super Lie algebra generators $K_{A B}$ (charges) such that
If follows from the supersymmetry algebra that $(E \delta_{A B} - K_{A B})$ 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 $(K_{A B})$ 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.
States with similar behaviour are also considered also in some models of soliton theory (English Wikipedia: Bogomol’nyi–Prasad–Sommerfield bound).
BPS states play a central role in the investigation of moduli spaces of classical vacua as they form 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.
In the context of supergravity BPS states correspond to super spacetimes admitting Killing vectors. These notably include extremal black brane solutions.
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 degeneracy of BPS states in string theory has been used to provide a microscopic interpretation of Bekenstein-Hawking entropy of black holes, see at black holes in string theory.
The following are some observations on the formalization of BPS states from the nPOV, in higher differential geometry, following (Sati-Schreiber 15).
Let $\mathbb{R}^{d-1,1|N}$ be a super-Minkowski spacetime, let $(d,N,p)$ be in the brane scan and write
for the correspoding super Lie algebra cocycle, as discussed at Green-Schwarz action functional, see (FSS 13) for the perspective invoked here.
Consider then $X$ a super-spacetime locally modeled on $\mathbb{R}^{d-1,1|N}$ as a Cartan geometry, solving the relevant supergravity equations of motion (e.g. 11-dimensional supergravity for $d= 11$, heterotic supergravity for $d = 10$ and $N = (1,0)$, type IIA supergravity for $d = 10$ and $N= (1,1)$ or type IIB supergravity for $d = 10$ $N= (2,0)$).
This means in particular that $X$ carries a super differential form
which is definite on $\phi$. This is the curvature of the WZW-term which defines the relevant super p-brane sigma-model with target space $X$.
By (AGIT 89) $X$ 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 $H^p_{dR}(X)$ 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 $\mathfrak{Pois}(X,\omega)$ induced by regarding $(X,\omega)$ as an pre-n-plectic supermanifold and restricting along the inclusion of the Killing vectors/Killing spinors into all the Hamiltonian vector fields.
(Here we are using that if an n-type is an extension of a 0-type, then its 0-truncation is still an extension by the 0-truncation of the original homotopy fiber.)
The elements in $H^p_{dR}(X)$ here are precisely the $p$-brane charges, as discussed in (AGIT 89, p. 8).
Hence $X$ is the more BPS the more odd-graded elements there are in $\tau_0 \mathfrak{Pois}(X,\omega)$ (or its restriction to super-isometries). Hence $X$ is a 1/2 BPS state of supergravity if the odd dimension of this is half that of $\mathbb{R}^{d-1,d|N}$, it is 1/4 BPS if the odd dimension is one fourth of that of $\mathbb{R}^{d-1,d|N}$, etc.
Notice that if
is a prequantization of $\omega$, i.e. an actual WZW term with curvature $\omega$, then $\mathfrak{Pois}(X,\omega)$ is supposed to be the Lie differentiation of the stabilizer group of $\mathbf{L}_{WZW}$, which is the quantomorphism n-group $QuantMorph(\mathbf{L}_{WZW})$. (This Lie differentiation statement is strictly shown only for $p = 0$ and $p = 1$ in dcct but clearly should hold generally.)
Hence we may regard $\mathbf{QuantMorph}(\mathbf{L}_{WZW})$ (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 $\mathbf{L}_{WZW}$ regraded as a local Lagrangian, and so this conceptually connects back to the considerations in (AGIT 89).
In 11-dimensional supergravity (M-theory) there are four kinds of 1/2 BPS states (the black M-branes) (e.g. Stelle 98, section 3 EHKNT 07):
The BPS bound derives its name from the discussion of magnetic monopoles in 4-dimensional Yang-Mills theory in
(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 extension of the term “BPS-saturated state” from this case to situations in string theory seems to have happened in
The original article identifying the role of BPS states in supersymmetric field theory is
Exposition and review includes
Andrew Neitzke, What is a BPS state?, 2012 (pdf)
Tudor Dimofte, Refined BPS invariants, Chern-Simons theory, and the quantum dilogarithm, 2010 (pdf, web)
Further developments are in
Jeffrey Harvey, Greg Moore, Algebras, BPS states, and strings, Nucl.Phys.B463:315–368 (1996) (doi); hep-th/9510182;
Jeffrey Harvey, Greg Moore, On the algebras of BPS states, Comm. Math. Phys. 197 (1998), 489–-519, doi, hep-th/9609017.
Ali Chamseddine, M. S. Volkov, Non-abelian BPS monopoles in $N=4$ gauged supergravity, Physical Review Letters 79: 3343–3346 (1997) hep-th/9707176.
Steven Weinberg, The quantum theory of fields, vol. II
Tudor Dimofte, Sergei Gukov, Refined, Motivic, and Quantum, arXiv:0904.1420
Davide Gaiotto, Gregory 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
Maxim Kontsevich, Yan Soibelman, Stability structures, motivic Donaldson-Thomas invariants and cluster transformations, arXiv:0811.2435
Maxim Kontsevich, Yan Soibelman, Motivic Donaldson-Thomas invariants: summary of results, arxiv/0910.4315
Dominic Joyce, Y. Song, A theory of generalized Donaldson-Thomas invariants, arxiv/0810.5645
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
Discussion of extremal/BPS black branes in supergravity (especially in 11-dimensional supergravity and 10d type II supergravity) includes
Kellogg Stelle, BPS Branes in Supergravity (arXiv:hep-th/9803116)
Jerome Gauntlett, Chris Hull, BPS States with Extra Supersymmetry, JHEP 0001 (2000) 004 (arXiv:hep-th/9909098)
Francois Englert, Laurent Houart, Axel Kleinschmidt, Hermann Nicolai, Nassiba Tabti, An $E_9$ multiplet of BPS states, JHEP 0705:065,2007 (arXiv:hep-th/0703285)
Andrew Callister, Douglas Smith, Topological BPS charges in 10 and 11-dimensional supergravity, Phys. Rev. D78:065042,2008 (arXiv:0712.3235)
Andrew Callister, Douglas Smith, Topological charges in $SL(2,\mathbb{R})$ covariant massive 11-dimensional and Type IIB SUGRA, Phys.Rev.D80:125035,2009 (arXiv:0907.3614)
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)
Specifically for $1/2^n$-BPS states of intersecting M-branes in 11d there is discussion in
see also
Discussion in the context of multiple M2-branes in the BLG model is in
Discussion for 4d supergravity, hence in KK-compactification of type II supergravity on a Calabi-Yau manifold is due to
Frederik Denef, Supergravity flows and D-brane stability, JHEP 0008:050,2000 (arXiv:hep-th/0005049)
Frederik Denef, Quantum Quivers and Hall/Hole Halos, JHEP 0210:023,2002 (arXiv:hep-th/0206072)
Discussion of more general classification of solutions to supergravity preserving some supersymmetry, i.e. admitting some Killing spinors includes
Jerome Gauntlett, Stathis Pakis, The Geometry of $D=11$ Killing Spinors, JHEP 0304 (2003) 039 (arXiv:hep-th/0212008)
Eric D'Hoker, John Estes, Michael Gutperle, Darya Krym, Paul Sorba, Half-BPS supergravity solutions and superalgebras, JHEP0812:047,2008 (arXiv:0810.1484)
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
reviewed in
This is for branes in the old brane scan (strings, membranes, NS5-branes), excluding D-branes and M5-brane.
The generalization oft this perspective to the M5-brane is discussed in
and the generalization to D-branes is discussed in
Detailed discussion of examples for various backgrounds is in
Discussion of this in higher differential geometry via the Poisson bracket Lie n-algebra is in
Discussion of relation of M5-brane BPS states to knot invariants includes
Edward Witten, Fivebranes and Knots (arXiv:1101.3216)
Sergei Gukov, Marko Stošić, Homological algebra of knots and BPS states (arXiv:1112.0030)
Ross Elliot, Sergei Gukov, Exceptional knot homology (arXiv:1505.01635)