Supersymmetry Across Dimensions, Signatures, and Reality Conditions

The symbol $\mathcal N$ does not have a dimension-independent numerical meaning. The invariant datum is the number of real supercharges together with their Lorentz representation, chirality, reality condition, and R-symmetry action. In Lorentzian signature the minimal package ranges from one real chiral charge in two dimensions to 32 real components in eleven dimensions; in Euclidean signature the compatible antilinear structures change, so analytic continuation usually requires complexifying the charges rather than erasing factors of $i$.

Required background. Clifford algebras and Spin groups supplies the mod-eight structure, while spinors, conjugations, and bilinears supplies chirality, charge conjugation, and real versus quaternionic modules.

Helpful background. Lorentz fields and Poincaré particles explains why field-spinor and one-particle representations are related but not identical.

Spinor data before supersymmetry notation

Fix a spacetime signature $(t,s)$, a Clifford relation, and an irreducible complex spinor module $S$. Four questions determine the supersymmetry notation:

1. Chirality: if $d=t+s$ is even, does $S=S_+\oplus S_-$, and may a charge occupy only one chirality?
2. Reality: is there an equivariant antilinear map $J:S\to S$ with $J^2=+1$? Its fixed points define a Majorana-type real form.
3. Quaternionic structure: if instead $J^2=-1$, a single fixed-point condition is impossible. A symplectic-Majorana condition pairs $S$ with an even-dimensional internal space carrying an antisymmetric form.
4. Compatibility: does $J$ preserve chirality or exchange $S_+$ and $S_-$? Only in the first case can a Weyl and Majorana-type condition be imposed simultaneously.

The complex Dirac dimension is $2^{\lfloor d/2\rfloor}$. A Weyl projection halves it in even dimension. A compatible Majorana condition halves the number of real components relative to an unconstrained complex spinor, while a symplectic condition acts on a doublet and removes the doubling it introduced. These statements are representation theory; a superalgebra additionally needs a symmetric spinor bilinear whose vector component can appear in $\{Q,Q\}$.

The mod-eight derivation and its bilinear symmetry conditions are given in Van Proeyen 1999, §§ 3–4 and Weinberg 2000, § 32.1 and Appendix 32.A, pp. 382–407. Nahm’s classification supplies the corresponding superalgebras Nahm 1978, pp. 149–166.

Minimal Lorentzian supercharges from two to eleven dimensions

The table uses one time direction and the site’s mostly-minus metric. “Minimal real $Q$” counts independent real operator components. The R-symmetry column gives the continuous zero-central-charge automorphism type for $N$ copies; a particular action may preserve only a subgroup, and nonzero central charges reduce it further.

$d$	Minimal spinor condition	Minimal real $Q$	Common extended notation	Zero-charge automorphism type
2	Majorana–Weyl, separately left and right	1 per chirality	$\mathcal N=(p,q)$	$O(p)\times O(q)$
3	Majorana	2	$\mathcal N=N$	$O(N)$
4	Majorana, equivalently one Weyl charge plus its adjoint	4	$\mathcal N=N$	$U(N)$
5	symplectic Majorana	8	minimal $\mathcal N=1$	$USp(2N)$
6	symplectic Majorana–Weyl	8 per chirality	$\mathcal N=(N_+,N_-)$	$USp(2N_+)\times USp(2N_-)$
7	symplectic Majorana	16	minimal $\mathcal N=1$	$USp(2N)$
8	Majorana; conjugation exchanges the two Weyl modules	16	minimal $\mathcal N=1$	$U(N)$
9	Majorana	16	$\mathcal N=N$	$O(N)$
10	Majorana–Weyl, separately for each chirality	16 per chirality	$\mathcal N=(N_+,N_-)$	$O(N_+)\times O(N_-)$
11	Majorana	32	minimal $\mathcal N=1$	$O(N)$

This is a spinor and algebra census, not a catalog of interacting theories. Requiring a massless multiplet with no spin above two gives the familiar ceiling of 32 real supercharges and $d\leq11$ under the usual finite-field-content assumptions; the state-counting argument appears in Weinberg 2000, § 32.2, pp. 393–397. It does not prove that every table entry has an interacting nongravitational realization.

Several common names are now transparent:

• four-dimensional $\mathcal N=1$ has $4$ real supercharges, and $\mathcal N=2$ has $8$;
• three-dimensional $\mathcal N=2$ has $4$ real supercharges, while $\mathcal N=4$ has $8$;
• five-dimensional $\mathcal N=1$ and six-dimensional $\mathcal N=(1,0)$ each have $8$;
• ten-dimensional type IIA has $(1,1)$ and type IIB has $(2,0)$ in one common chirality convention; both have $32$ real supercharges.

Thus “same $\mathcal N$” does not mean “same amount of supersymmetry,” and “same number of real supercharges” does not mean “same algebra.”

R-symmetry as the commutant of the spinor real form

Let $Q^I$ denote copies of a minimal spinor. The transformations on $I$ that preserve the momentum term in $\{Q,Q\}$ form the algebraic R-symmetry. Its compact type follows from the commutant of the spin representation:

$$\begin{array}{c|c} \text{spinor type} & \text{multiplicity-space symmetry}\\ \hline \text{real} & O(N)\\ \text{complex} & U(N)\\ \text{quaternionic} & USp(2N) \end{array}$$

In even dimensions, inequivalent chiralities can carry independent multiplicities, giving a product group. This rule explains rather than memorizes the last column of the table. It also distinguishes an R-symmetry from flavor: an R-generator acts nontrivially on $Q$, whereas a flavor generator commutes with every supercharge.

Central charges transform in tensor representations of this multiplicity space. Selecting a charge sector retains only the stabilizer of the central-charge matrix. Therefore the automorphism group of the abstract zero-charge algebra, the symmetry of a Lagrangian, the anomaly-free quantum R-symmetry, and the stabilizer of a state need not coincide. Strathdee’s cross-dimensional representation catalog makes these qualifications explicit in Strathdee 1987, pp. 273–300.

Positive-definite Euclidean signature

For $Spin(d)$, the type of an irreducible spinor is periodic modulo eight. The following table records the representation-theoretic reality structure; it does not by itself choose a reflection-positive Euclidean QFT contour.

$d\bmod 8$	Spinor type	Action of conjugation	Consequence
0	real chiral	preserves each $S_\pm$, $J^2=+1$	a real chiral condition is possible at the module level
1, 7	real nonchiral	$J^2=+1$	a Majorana-type real form is possible
2, 6	complex chiral	exchanges $S_+$ and $S_-$	a single chiral spinor has no real form
3, 5	quaternionic nonchiral	$J^2=-1$	a symplectic doublet is needed for a real condition
4	quaternionic chiral	preserves chirality but has $J^2=-1$	a symplectic-Weyl doublet is possible

The four-dimensional warning is especially important. Lorentzian $Spin(1,3)\simeq SL(2,\mathbb C)$ has left and right Weyl modules related by Hermitian conjugation. Euclidean $Spin(4)\simeq SU(2)_L\times SU(2)_R$ has two independent pseudoreal doublets. Consequently a Lorentzian $\mathcal N=1$ pair

$$Q_\alpha,\qquad \bar Q_{\dot\alpha}=(Q_\alpha)^\dagger$$

continues to independent complex Euclidean generators

$$Q_\alpha,\qquad \widetilde Q_{\dot\alpha},$$

not to two variables related by pointwise Hermitian conjugation. A Euclidean functional integral must state its integration cycle or reflection operation separately. Counting the complexified components as new physical supercharges would incorrectly relabel the theory.

A checked cross-dimensional translation

Compare four-dimensional $\mathcal N=2$ with three-dimensional $\mathcal N=4$.

In four dimensions,

$$Q^I_\alpha,\qquad I=1,2,\qquad \alpha=1,2$$

contains $2\times2$ complex components constrained by the adjoint relation to $\bar Q_{\dot\alpha I}$: eight real charges. After reduction on one spatial direction, a four-dimensional Weyl spinor becomes a three-dimensional complex two-component spinor, equivalently two Majorana spinors. The index set therefore reorganizes into four three-dimensional Majorana charges $Q^A_a$, again $4\times2=8$ real components.

The invariant checkpoint is the rank of the rest-frame positive matrix $\{Q,Q^\dagger\}$: it is eight over the reals on both sides. What changes is the Lorentz group and R-symmetry. The four-dimensional zero-charge automorphism $U(2)$ embeds in the three-dimensional $SO(4)\simeq SU(2)\times SU(2)$ R-symmetry because the commutant of the reduced Lorentz action on the supercharges is larger. In field realizations, the extra symmetry also reorganizes the scalar descending from a higher-dimensional gauge field. Equality of real counts therefore licenses dimensional reduction, not an identification of the two algebras.

Common pitfalls

Counting a Weyl spinor as two real charges in four dimensions. A two-component Weyl charge is complex and comes with its Hermitian adjoint; four-dimensional $\mathcal N=1$ has four real supercharges.

Using Lorentzian conjugation after Wick rotation. In four Euclidean dimensions, dotted and undotted charges are independent complex variables until a contour or reflection structure is supplied.

Reading the table as an existence theorem. The Clifford module determines candidate algebraic data. Interacting QFT existence, unitarity, anomaly cancellation, and a spin ceiling are additional requirements.

Check your understanding

How many real supercharges are carried by four-dimensional $\mathcal N=4$, three-dimensional $\mathcal N=8$, and six-dimensional $\mathcal N=(2,0)$?

Answer

All three have 16 real supercharges: $4\times4$ in four dimensions, $8\times2$ in three dimensions, and $2\times8$ chiral charges in six dimensions. Their Lorentz spinors, R-symmetries, and possible central or tensorial extensions are nevertheless different.

References

• Werner Nahm, “Supersymmetries and Their Representations,” Nuclear Physics B 135 (1978), 149–166, DOI.
• Jonathan Strathdee, “Extended Poincaré Supersymmetry,” International Journal of Modern Physics A 2 (1987), 273–300, DOI.
• Antoine Van Proeyen, “Tools for Supersymmetry,” in Annals of the University of Craiova, Physics 9 (1999), 1–48, arXiv.
• Steven Weinberg, The Quantum Theory of Fields, Volume III: Supersymmetry, Cambridge University Press (2000), Chapter 32, DOI.