Extended Supersymmetry, R-Symmetry, and Central Charges

Extended supersymmetry adds a multiplicity index $I$ to the supercharges. The mixed anticommutator still produces momentum, while the same-chirality anticommutator can contain an antisymmetric complex matrix of scalar central charges. R-symmetry rotates the supercharges and therefore acts on that matrix by unitary congruence. Tensorial charges are a distinct extension: they commute with translations but transform under Lorentz transformations and typically measure extended objects or boundary sectors.

Required background. The four-dimensional N=1 algebra fixes the two-component convention. Projective actions and central extensions supplies the distinction between a central operator and a projective phase.

Helpful background. Disorder operators and singular boundary conditions gives examples in which a charge is defined by asymptotic or defect data rather than a local particle density.

The extended four-dimensional algebra

Let $I,J=1,\ldots,\mathcal N$. In the conventions of the preceding page,

$$\begin{aligned} \{Q^I_\alpha,\bar Q_{\dot\beta J}\} &=2\sigma^\mu_{\alpha\dot\beta}P_\mu\,\delta^I{}_J,\\ \{Q^I_\alpha,Q^J_\beta\} &=2\epsilon_{\alpha\beta}Z^{IJ},\\ \{\bar Q_{\dot\alpha I},\bar Q_{\dot\beta J}\} &=2\epsilon_{\dot\alpha\dot\beta}\bar Z_{IJ},\\ [P_\mu,Q^I_\alpha]&=0. \end{aligned}$$

Hermitian conjugation gives $(Z^{IJ})^\dagger=\bar Z_{IJ}$. Exchanging the complete pairs $(I,\alpha)$ and $(J,\beta)$ leaves the anticommutator unchanged but changes the sign of $\epsilon_{\alpha\beta}$, so

$$Z^{IJ}=-Z^{JI}.$$

This is why a point-particle scalar central charge vanishes for $\mathcal N=1$ and first appears for $\mathcal N=2$. The graded Jacobi identities show that $Z^{IJ}$ commutes with translations and supercharges, and Lorentz covariance makes it a scalar. An even internal generator can rotate the matrix $Z^{IJ}$; only its stabilizer remains a symmetry in a fixed charge sector. Thus “central charge” here means central in the supertranslation and Poincaré sector, while a generator central in the entire enlarged algebra must also be invariant under every internal action. The derivation is given in Weinberg 2000, § 25.2, pp. 30–38 and the original classification in Haag, Łopuszański, and Sohnius 1975, pp. 257–274.

Nor does a commuting charge have to take the same value on the entire Hilbert space. In a fixed irreducible sector of the algebra that commutes with it, Schur’s lemma makes it a complex number $z^{IJ}$; different superselection sectors may carry different eigenvalues.

R-symmetry and the central-charge stabilizer

When $Z=0$, the algebra is invariant under

$$Q^I_\alpha\longmapsto U^I{}_JQ^J_\alpha, \qquad U\in U(\mathcal N).$$

This is the algebraic R-symmetry. If $Z$ is retained as an operator-valued tensor, it transforms as

$$Z\longmapsto UZU^{\mathsf T}.$$

A fixed charge sector therefore preserves only

$$G_Z=\{U\in U(\mathcal N):UZU^{\mathsf T}=Z\}.$$

The same statement follows from Jacobi identities. If an internal generator $T_A$ acts by

$$[T_A,Q^I_\alpha]=(t_A)^I{}_JQ^J_\alpha,$$

then

$$[T_A,Z]=t_AZ+Zt_A^{\mathsf T}.$$

An ordinary flavor generator has $t_A=0$ and commutes with $Q$; an R-generator has $t_A\neq0$. The automorphism of the abstract algebra, the symmetry of an action, the anomaly-free quantum symmetry, and the subgroup preserved by a state are therefore four different objects.

For $\mathcal N=2$,

$$Z^{IJ}=\epsilon^{IJ}Z.$$

The identity $U\epsilon U^{\mathsf T}=(\det U)\epsilon$ shows that $SU(2)_R$ preserves a fixed nonzero $Z$, while the overall $U(1)$ rotates its phase. That $U(1)$ may still be used spurionically if couplings or charges transform, but it is not a symmetry within one fixed nonzero-$Z$ sector.

Canonical central-charge blocks

Every complex antisymmetric matrix admits a unitary skew-normal form,

$$UZU^{\mathsf T} =\operatorname{diag}(z_1\epsilon,z_2\epsilon,\ldots,z_r\epsilon,0), \qquad \epsilon=\begin{pmatrix}0&1\\-1&0\end{pmatrix},$$

where phases may be chosen so that $z_a\geq0$. Equivalently, each nonzero singular value of $Z$ occurs twice. This is the correct basis for the positivity problem because the supercharge anticommutator decomposes into independent $2\times2$ internal blocks.

For $\mathcal N=2$, there is one block. In a massive rest frame, identify dotted and undotted $SU(2)$ little-group indices and rephase the charges so $Z=|Z|$. Suitable combinations of $Q^1$ and $(Q^2)^\dagger$ have anticommutators proportional to

$$2(M+|Z|), \qquad 2(M-|Z|).$$

Positivity therefore singles out the invariant magnitude $|Z|$, not its convention-dependent phase, as the particle charge entering the BPS inequality. The complete diagonalization, state-count reduction, and recombination limit are derived on BPS bounds and shortening.

As a convention check, the eigenvalues of $ZZ^\dagger$ are basis invariant. Any proposed R-rotation or phase redefinition must preserve them, and hence preserve every mass bound.

Scalar central charges and tensorial charges

A scalar $Z$ commutes with Lorentz transformations. Charges carried by strings, walls, and higher-dimensional objects need not. The general supertranslation algebra can contain bilinears

$$\{Q,Q\} =(C\Gamma^\mu)P_\mu +\sum_p\frac1{p!}(C\Gamma^{\mu_1\cdots\mu_p}) Z_{\mu_1\cdots\mu_p},$$

but only those values of $p$ for which the spinor matrix is symmetric are allowed. The permitted ranks depend on dimension, signature, chirality, and reality. Each $Z_{(p)}$ must also pass the graded Jacobi identities.

The logical distinction is:

Charge	Translation bracket	Lorentz behavior	Typical physical datum
scalar central $Z$	$[P,Z]=0$	scalar; commutes with $M_{\mu\nu}$	point-particle electric, magnetic, or topological sector
tensorial $Z_{\mu_1\cdots\mu_p}$	$[P,Z_{(p)}]=0$	transforms as a $p$-form	oriented string, wall, brane, boundary, or winding charge

In four dimensions, for example, a symmetric spinor tensor $(\sigma^{\mu\nu})_{\alpha\beta}Z_{\mu\nu}$ can occur in $\{Q_\alpha,Q_\beta\}$ when wall or boundary sectors invalidate the point-particle assumptions. In eleven dimensions the standard schematic extension is

$$\{Q,Q\} =(C\Gamma^\mu)P_\mu +\frac12(C\Gamma^{\mu\nu})Z_{\mu\nu} +\frac1{5!}(C\Gamma^{\mu_1\cdots\mu_5})Z_{\mu_1\cdots\mu_5}.$$

The two- and five-form terms are associated with extended-object sectors. They are sometimes called “central charges of the supertranslation algebra,” but they are not central in the full Lorentz algebra. The dimension-dependent construction and the point-particle loophole are detailed in Weinberg 2000, § 32.3, pp. 397–401.

Such charges are often surface terms. Their value depends on orientation, normalization, asymptotic fields, topology, and boundary conditions. The algebra can imply an energy bound in a specified sector; it does not prove that a soliton exists, is stable, or survives quantum corrections. Witten and Olive established the central-charge relation to topological monopole sectors in Witten and Olive 1978, pp. 97–101.

A charge-record checklist

Before using an extended bracket, record:

1. spacetime dimension, signature, spinor real form, and chirality;
2. the number of independent real supercharges;
3. the adjoint and normalization of $Q$, $P$, and every $Z$;
4. whether a charge is a Lorentz scalar or a tensorial surface charge;
5. its transformation under the algebraic R-symmetry;
6. the stabilizer of the chosen charge sector;
7. the global charge lattice, orientation, and boundary conditions when physical charges are claimed; and
8. the invariant check—normally singular values of $Z$, a Dirac pairing, or a positive anticommutator eigenvalue.

Common pitfalls

Calling every commuting surface term central. Commuting with translations is not enough. A tensorial charge fails to commute with Lorentz generators and is not central in the full super-Poincaré algebra.

Using $U(\mathcal N)$ after fixing a generic $Z$. The zero-charge algebra has $U(\mathcal N)$ automorphisms. A nonzero central-charge matrix retains only its unitary-congruence stabilizer.

Turning a BPS bound into an existence claim. Positivity says what the mass must obey if a state with those charges exists. Existence, stability, and wall crossing belong to the dynamics of the relevant charge sector.

Check your understanding

Why is $Z^{IJ}$ antisymmetric, and why does that eliminate a scalar point-particle central charge for $\mathcal N=1$?

Answer

The anticommutator is symmetric under $(I,\alpha)\leftrightarrow(J,\beta)$, while $\epsilon_{\alpha\beta}$ is antisymmetric. Therefore its coefficient must also be antisymmetric: $Z^{IJ}=-Z^{JI}$. A $1\times1$ antisymmetric matrix vanishes.

References

• Rudolf Haag, Jan T. Łopuszański, and Martin Sohnius, “All Possible Generators of Supersymmetries of the S-Matrix,” Nuclear Physics B 88 (1975), 257–274, DOI.
• Steven Weinberg, The Quantum Theory of Fields, Volume III: Supersymmetry, Cambridge University Press (2000), §§ 25.2 and 32.3, DOI.
• Edward Witten and David I. Olive, “Supersymmetry Algebras That Include Topological Charges,” Physics Letters B 78 (1978), 97–101, DOI.