Supersymmetry Algebras and BPS Bounds

Supersymmetric black holes are special because their mass is constrained by a charge appearing inside the supersymmetry algebra. Positivity turns that algebraic fact into a sharp inequality; saturation removes supercharges, shortens the state’s multiplet, and makes selected quantities insensitive to many continuous deformations. This page proves the bound and states exactly what saturation does—and does not—protect.

Required background. Gauge Theory and String-Theory Preparation supplies conserved charges, spinors, and supersymmetry notation. Gravitational Actions and Boundary Terms supplies the Hamiltonian and asymptotic-boundary setting in which ADM energy and charges are defined.

Helpful background. Quantum Gravity gives the effective-theory setting, Quantum Information clarifies protected subspaces and traces, and Black-Hole Geometry and Mechanics supplies extremal horizons and thermodynamic charges.

Central charges in the supersymmetry algebra

For a four-dimensional theory with an asymptotically Poincaré-invariant vacuum and extended supersymmetry, the massive rest-frame algebra can be organized as

$$\{Q^A_\alpha,(Q^B_\beta)^\dagger\} =2M\,\delta^{AB}\delta_{\alpha\beta}, \qquad \{Q^A_\alpha,Q^B_\beta\} =2\epsilon_{\alpha\beta}Z^{AB}.$$

Here $A,B$ are R-symmetry labels and the antisymmetric matrix $Z^{AB}$ is a central charge. In a gravitational theory $M$ is the energy measured at the chosen asymptotic boundary. Electric, magnetic, brane, and momentum charges can enter $Z^{AB}$; the precise linear combination depends on the compactification and scalar values at infinity. Peet derives the corresponding charge contribution and mass bound for string-theory branes in Peet 2000, §2.3, pp. 19–21, Eqs. (2.24)–(2.32).

A unitary change of R-symmetry basis puts $Z^{AB}$ into skew-diagonal $2\times2$ blocks with complex eigenvalues $z_i$. For each block, a phase rotation makes $z_i$ real and nonnegative. The algebra then reduces to positive operators whose eigenvalues are proportional to $M\pm |z_i|$.

BPS bound: theorem and hypotheses

Theorem (supersymmetric positivity bound). Let a quantum theory have a positive-definite physical Hilbert-space inner product, self-adjoint energy, well-defined asymptotic conserved charges, and an unbroken extended supersymmetry algebra with central-charge eigenvalues $z_i$. Every normalizable state of fixed charges obeys

$$M\geq \max_i |z_i|.$$

If equality holds for one or more $i$, a nonzero linear combination of supercharges annihilates the state. Conversely, if such a combination and its adjoint annihilate the state and the algebra has no omitted positive charge terms, the corresponding bound is saturated.

The hypotheses matter. Gauge constraints must already have been imposed so that negative-norm states are absent. Surface terms must be finite and included in the charge algebra. The statement is sector-by-sector: it compares states with fixed asymptotic charges and fixed boundary conditions. It is not a claim that the numerical central charge is independent of moduli.

Proof architecture

Choose a skew-eigenvalue $z=|z|e^{i\varphi}$, with $Z^{12}=z$ and $\epsilon_{12}=1$. In this block an explicit pair of combinations is

$$A=\frac12\left(Q^1_1-e^{i\varphi}(Q^2_2)^\dagger\right), \qquad B=\frac12\left(Q^1_1+e^{i\varphi}(Q^2_2)^\dagger\right).$$

The two diagonal terms each contribute $2M$, and the two mixed terms each contribute $2|z|$ with the chosen sign. Thus the displayed normalization gives exactly

$$\{A,A^\dagger\}=M-|z|, \qquad \{B,B^\dagger\}=M+|z|.$$

For any physical state $|\psi\rangle$,

$$\langle\psi|\{A,A^\dagger\}|\psi\rangle =\lVert A|\psi\rangle\rVert^2 +\lVert A^\dagger|\psi\rangle\rVert^2\geq0.$$

Therefore $M-|z|\geq0$. Repeating the argument for every skew block proves the bound. If $M=|z|$, the two norms in the last line vanish separately, so both $A$ and $A^\dagger$ annihilate the state. Those missing raising and lowering operations shorten the representation. This positivity mechanism is the central argument of Witten and Olive 1978, pp. 97–101; its gravitational surface-charge realization is developed in Gibbons and Hull 1982, pp. 190–194.

With several saturated eigenvalues, more independent combinations vanish. The fraction of preserved supersymmetry is the number of real annihilating supercharges divided by the number in the asymptotic vacuum algebra. Thus “BPS” alone does not specify whether a solution is half-, quarter-, or one-eighth-BPS.

Short multiplets and protected information

A generic massive state is in a long multiplet: all fermionic creation operators generated by the supercharges act nontrivially. Saturation removes some of them, so a short multiplet contains fewer states. A short multiplet cannot vary continuously into a long one unless it meets other short multiplets with compatible charges and quantum numbers. This recombination obstruction is the origin of protection.

What is protected is representation-theoretic data—often a supersymmetric index or a helicity supertrace—not automatically the raw number of states. The mass of a BPS state tracks $|Z(\Gamma,t_\infty)|$, which can vary with asymptotic moduli $t_\infty$ even though the equality $M=|Z|$ persists. Multi-center bound states can also appear or disappear across walls of marginal stability. The microscopic-index discussion separates this invariant signed count from an absolute degeneracy.

For a regular supersymmetric black hole, the Killing-spinor equation supplies the classical counterpart: an asymptotic supercharge annihilates the solution, and the ADM mass equals the appropriate charge norm. Saturation often implies extremality, but the logical statements are distinct. Extremality is a degeneracy of the horizon or a zero-temperature condition; BPS saturation is a statement about the supersymmetry algebra. Extremal non-BPS black holes exist.

Checks and failure modes

Three checks prevent common overstatements. First, every term in the charge algebra must have the same mass dimension and the same asymptotic normalization. Second, the preserved supersymmetries found from the Killing-spinor equations must agree with the number of zero anticommutators. Third, the solution’s ADM mass and Page or electromagnetic charges must be evaluated in the same frame and boundary convention.

The proof fails if the physical inner product is indefinite, supersymmetry is explicitly or anomalously broken, boundary charges are ill-defined, or the chosen state is not in the domain of the supercharges. It also does not prove horizon regularity. A singular brane configuration can saturate an algebraic bound, while a regularity theorem requires separate geometric hypotheses. Nor does a BPS equality alone determine an entropy: one still needs a protected count, a horizon calculation, or both.

Exercises

Show that a theory with two skew central-charge eigenvalues $|z_1|\gt|z_2|$ and mass $M=|z_1|$ saturates only the first bound, and explain why the second block still generates a long pair of fermionic directions.

Solution

Diagonalization gives positive anticommutators proportional to $M-|z_i|$ and $M+|z_i|$ for each block. For $i=1$, $M-|z_1|=0$, so the corresponding operator and its adjoint have zero norm on the state and annihilate it. For $i=2$, $M-|z_2|=|z_1|-|z_2|\gt0$; hence that norm need not vanish, and its raising operation remains active. The preserved fraction is therefore set by the first block only. If $|z_1|=|z_2|$, the conclusion changes because both zero anticommutators can occur.

References

• Gibbons, Gary W., and Christopher M. Hull. “A Bogomolny Bound for General Relativity and Solitons in N=2 Supergravity.” Physics Letters B 109 (1982): 190–194. DOI.
• Peet, Amanda W. “TASI Lectures on Black Holes in String Theory.” arXiv:hep-th/0008241 (2000). Open manuscript.
• Witten, Edward, and David Olive. “Supersymmetry Algebras That Include Topological Charges.” Physics Letters B 78 (1978): 97–101. DOI.