SUSY Algebras and Unitary Representations

A supersymmetric structure is not specified by writing an anticommutator of supercharges. It must pass five linked tests: its graded brackets must satisfy the Jacobi identities; its supercharges must obey a spinor reality condition available in the chosen dimension and signature; its positive-energy representations must be unitary; any shortening must follow from null supercharge combinations at a sharp bound; and the data passed to amplitudes or conformal field theory must be separated from information that only dynamics can supply. This chapter develops those tests as one reusable method.

Helpful background. The chapter uses spinor conjugations, invariant bilinears, and Fierz rearrangements to decide which anticommutators and reality conditions exist; Lorentz, field, and Poincaré representations to distinguish covariant fields from one-particle states; and multiplets, invariants, and selection rules to organize symmetry representations. If any of those distinctions is unfamiliar, use the preparation check below before beginning the derivations.

The five consistency tests

The Lorentzian unitary core is a real Lie superalgebra represented on a positive physical Hilbert space. Euclidean continuation instead begins with its complexification and requires a separately specified reflection operation or integration cycle. In either description, the even part contains spacetime symmetry and may contain internal symmetry, while the odd part is generated by spinorial operators $Q$. The algebra is only the beginning. The following questions locate most errors encountered when moving between dimensions, signatures, representations, and applications.

Test	Question to answer	Typical failure	Where it is resolved
Graded closure	Do all even–even, even–odd, and odd–odd brackets obey the graded Jacobi identity under the hypotheses being used?	A proposed mixed spacetime/internal generator contradicts Lorentz covariance or an omitted hypothesis invalidates a no-go conclusion.	Graded spacetime symmetry
Reality	Does the chosen spinor module admit the stated Majorana, Weyl, symplectic, or complex structure in this signature?	The symbol $N$ is copied across dimensions even though the number of real supercharges has changed.	Dimensions and reality conditions
Unitarity	Is the odd anticommutator a positive-semidefinite operator on physical states, and has the little-group representation been completed under CPT when necessary?	A covariant field count is mistaken for a physical-state count, or negative-norm states are retained.	Unitary supermultiplets
Shortening	Which eigenvalues of the positive anticommutator vanish at the bound, and which descendants become null?	A small multiplet is called BPS without identifying the relevant scalar or tensorial charge term, the saturated eigenvalue inequality, and the resulting null quotient.	BPS bounds and recombination
Interface	Which representation-theoretic data survive in an on-shell or superconformal description, and what remains dynamical?	A Ward identity or shortening condition is treated as if it determined an amplitude or a CFT correlator.	On-shell Ward identities and the CFT handoff

These tests are coupled. The admissible real spinor fixes the number and type of odd generators. In a fixed Lorentzian momentum-and-charge sector, the rank of the positive anticommutator fixes the number of active fermionic oscillators. The Clifford-vacuum representation and any separate CPT completion then fix the total multiplet size, while rank loss produces null directions. In radial quantization, the analogous $Q$–$S$ Gram matrices govern superconformal shortening. The surviving quantum numbers then become input to amplitude or conformal-representation calculations.

A route through the chapter

The pages are ordered so that each new structure is justified before it is used.

1. Graded Spacetime Symmetry and the Supersymmetry Theorems states the Coleman–Mandula and Haag–Łopuszański–Sohnius hypotheses, derives the allowed graded form, and identifies genuine loopholes rather than treating a theorem slogan as universal.
2. Supersymmetry Across Dimensions, Signatures, and Reality Conditions turns the Clifford module, its real structure, and its commutant into a checked count of real supercharges and the corresponding R-symmetry type.
3. The Four-Dimensional N=1 Super-Poincaré Algebra fixes one complete two-component convention, including index motion and adjoints, and verifies representative Jacobi identities, positive energy, and closure onto translations.
4. Extended Supersymmetry, R-Symmetry, and Central Charges adds multiple supercharges and separates four notions that are often conflated: the zero-charge R-automorphism, its congruence action on a family of central-charge matrices, the stabilizer of a fixed matrix, and genuine Haag–Łopuszański–Sohnius internal generators that commute with scalar central charges. It then distinguishes scalar central extensions from Lorentz-tensorial charges.
5. Massive and Massless Unitary Supermultiplets converts the positive anticommutator into fermionic oscillators in massive and null momentum frames, counts states, and explains when CPT adds a conjugate multiplet.
6. BPS Bounds, Shortening, and Multiplet Recombination diagonalizes the centrally extended algebra, obtains the BPS inequality from positivity, quotients null descendants, and tracks how short multiplets join away from the bound.
7. On-Shell Supermultiplets and Supersymmetric Ward Identities packages all-outgoing dual external wavefunctions into Grassmann polynomials, separates CPT conjugation from Grassmann Fourier transformation, and derives both the generic rank-two Ward solution and the exceptional three-point rank-one invariant.
8. Superconformal Algebras, Shortening Data, and the CFT Handoff relates Poincaré and conformal supercharges, derives a representative first-level norm and the complete four-dimensional N=1 scalar branches, gives exact recombination and normalization-transfer data, and separates algebraic input from conformal dynamics.

For a first pass, follow the list in order. For a targeted calculation, start with the reality-condition page and specialize the algebra. Use the unitary-representation construction before on-shell packaging. For central-charge shortening, combine the extended algebra with the unitary construction before the BPS analysis; the superconformal handoff then reuses the positive-norm, null-quotient, and recombination logic in radial quantization.

Conventions that connect the calculations

The site-wide metric is $(+---)$, so a massive momentum has $p^2=m^2>0$ and a positive-energy rest frame has $p^\mu=(m,0,0,0)$. Hermitian conjugation relates $Q_\alpha$ to $\bar Q_{\dot\alpha}$ in Lorentzian four-dimensional examples. The odd bracket is an anticommutator, and Hilbert-space positivity is used in the form

$$\lVert A\lvert\psi\rangle\rVert^2 +\lVert A^\dagger\lvert\psi\rangle\rVert^2 =\langle\psi\rvert\{A^\dagger,A\}\lvert\psi\rangle\ge 0.$$

Each detailed page states the additional spinor, phase, and normalization choices needed for its derivation. In particular:

• $N$ counts copies of a specified spinor type; it is not a dimension-independent count of real components.
• The zero-charge algebra has an R-automorphism group acting nontrivially on $Q$. A supercharge-basis change carries $Z$ by congruence, and a fixed numerical $Z$ retains only its stabilizer; an action, anomaly, or vacuum may reduce the physical symmetry further. Under the Haag–Łopuszański–Sohnius particle-S-matrix hypotheses, genuine internal generators commute with scalar central charges. An explicitly adjoined R-derivation that moves $Z$ defines a different enlarged algebra.
• Higher-rank tensorial charges may commute with translations in the bulk supertranslation algebra, but they transform under Lorentz symmetry and therefore are not central in the full super-Poincaré algebra.
• Multiplet state counts refer to the positive-definite physical Hilbert space. Gauge-redundant field components and null descendants are removed before counting.
• A BPS fraction is meaningful only after specifying the real supercharge count and the number of independent real combinations that annihilate the state.

These statements prevent the most common translation error: carrying a formula into a new dimension or signature while silently keeping the old meaning of conjugation, $N$, or positivity.

Check your preparation

You are ready for the chapter if you can perform the following short checks.

Spinor check. In four-dimensional Lorentzian signature, explain why complex conjugation exchanges the $(\tfrac12,0)$ and $(0,\tfrac12)$ Weyl representations rather than defining a reality condition on either representation alone.

Answer and route

The two chiral Weyl modules are inequivalent complex representations of $\operatorname{Spin}(1,3)\cong SL(2,\mathbb C)$, and complex conjugation maps one to the other. A Majorana spinor therefore pairs the two chiralities. If this is unfamiliar, review spinor conjugations and bilinears before the dimension table.

One-particle check. For a massless four-dimensional particle, identify the subgroup of the Poincaré group that labels helicity states and explain why an off-shell vector field has more components than a physical helicity representation.

Answer and route

The little group is $ISO(2)$; ordinary finite-helicity particles use representations on which its translation subgroup acts trivially, leaving the $SO(2)$ helicity. Gauge redundancy and equations of motion remove the unphysical components of a covariant vector field. Review Poincaré particle representations if those reductions are not yet automatic.

Algebra check. State the graded Jacobi identity for three homogeneous elements and identify the sign that differs from an ordinary Lie algebra.

Answer and route

For degrees $|X|,|Y|,|Z|\in\{0,1\}$,

$$(-1)^{|X||Z|}[X,[Y,Z\}\} +(-1)^{|Y||X|}[Y,[Z,X\}\} +(-1)^{|Z||Y|}[Z,[X,Y\}\}=0.$$

The degree-dependent signs implement antisymmetry in the graded sense; in particular, the bracket of two odd elements is symmetric. Begin with graded spacetime extensions if you want to rebuild this carefully.

Review the chapter

The following prompts test more than formula recall. A satisfactory answer should name its assumptions, carry the relevant normalization consistently, and say what would invalidate the conclusion.

1. Retrieve the four-dimensional algebra. Write the nonzero odd anticommutator of four-dimensional $N=1$ super-Poincaré symmetry, its Hermitian conjugation rule, and the Lorentz transformation law of $Q_\alpha$.

Answer criteria

Use one internally consistent two-component convention and obtain $\{Q_\alpha,\bar Q_{\dot\beta}\}=2\sigma^\mu_{\alpha\dot\beta}P_\mu$, with $(Q_\alpha)^\dagger=\bar Q_{\dot\alpha}$. In the minimal point-particle algebra, $\{Q,Q\}=\{\bar Q,\bar Q\}=0$; Lorentz-tensor surface extensions in wall or boundary sectors are outside this specialization. The Lorentz bracket must place $Q$ in $(\tfrac12,0)$ and $\bar Q$ in $(0,\tfrac12)$. Repair signs and factors on the N=1 algebra page.

2. Explain a theorem boundary. A two-dimensional integrable model has infinitely many conserved charges. Does this contradict Coleman–Mandula or the supersymmetric extension theorem?

Answer criteria

No. The original Coleman–Mandula and Haag–Łopuszański–Sohnius results used here are four-dimensional particle-S-matrix theorems, and the complete Haag–Łopuszański–Sohnius analysis further assumes a massive setting with no long-range forces. A two-dimensional integrable model lies outside that setting and also has exceptional scattering kinematics that defeat the generic-scattering analytic-continuation step. No contradiction follows. Use the theorem-assumption analysis to identify the relevant premises precisely.

3. Derive positivity. Starting from the $N=1$ odd anticommutator, show that the expectation value of $P^0$ is nonnegative in a unitary representation.

Answer criteria

Choose a commuting spinor $\xi$ and set $A=\xi^\alpha Q_\alpha$. Positivity of $\{A,A^\dagger\}$ gives a nonnegative contraction of $P_\mu$ with the future-directed null vector $\xi\sigma^\mu\xi^\dagger$. Summing over an orthonormal spinor basis isolates a positive multiple of $P^0$. State the spectrum and adjoint assumptions; without a positive-definite Hilbert space, the inference fails.

4. Translate between dimensions. Compare four-dimensional $N=2$ with three-dimensional $N=4$. What is invariant, and what notation changes?

Answer criteria

Both have eight real Poincaré supercharges, but their Lorentz spinors and R-symmetry presentations differ: four-dimensional chirality disappears, and the zero-charge $U(2)$ automorphism embeds in the enhanced three-dimensional $SO(4)$. If nonzero Kaluza–Klein momentum is retained, the reduced component $P_3$ is a three-dimensional Lorentz scalar and may act as a central charge; strict zero-mode reduction has $P_3=0$. Check the full translation on dimensions and reality conditions.

5. Compare massive and massless multiplets. Why does a null momentum reduce the number of active fermionic oscillators, and how does this affect the helicity span?

Answer criteria

For null $p$, the matrix $p_\mu\sigma^\mu$ has rank one rather than rank two. The zero-eigenvalue supercharge acts trivially in a unitary representation, leaving half as many creation operators as in the massive rest frame. The remaining operators shift helicity by one half; the unitary-multiplet construction also shows when CPT completion is needed.

6. Diagnose a shortening claim. A state in a massive four-dimensional $N=2$ charge sector with $Z\neq0$ has $M=\lvert Z\rvert$. What else must be checked before calling its multiplet BPS?

Answer criteria

Fix the central-charge and supercharge normalization, diagonalize the full positive odd anticommutator in the rest frame, verify that the state carries the stated $Z$, and identify the null supercharge combinations. The short representation is the quotient by their null descendants. The nonzero hypotheses matter: $M=\lvert Z\rvert=0$ belongs to the massless rank-loss problem, not this massive rest-frame derivation. The equality alone is ambiguous without these data; the BPS derivation supplies the complete test.

7. Transfer the algebra on shell. At generic rank-two kinematics, explain why a factor $\delta^{(2N)}(\mathsf q)$ solves only the Ward-identity part of a supersymmetric amplitude problem, and state what changes on the rank-one three-point branch.

Answer criteria

Write the full amplitude as $\mathbf A_n=(2\pi)^4\delta^{(4)}(P)\mathscr A_n$ and impose the supersymmetry Ward identities on $P=0$. At generic rank-two kinematics, $\delta^{(2N)}(\mathsf q)$ is annihilated by the multiplication charges, while momentum conservation makes the differentiation charges annihilate it. Supersymmetry still leaves coefficient functions and, at higher Grassmann degree, additional invariant polynomials undetermined; little-group covariance, internal charges, locality, factorization, mass dimension, and analytic structure supply further constraints. On the square-bracket three-point branch, $\delta^{(2N)}(\mathsf q)$ vanishes and a different rank-one invariant is required. See on-shell Ward identities for the operator derivation.

8. Separate symmetry from dynamics. Given a superconformal primary, list the data determined by the algebra and the additional data needed to compute a four-point function.

Answer criteria

The algebra fixes admissible Lorentz and R-symmetry labels, unitarity inequalities, shortening equations, descendant relations, recombination rules, and selection rules; it also constrains correlator tensor structures. A specified four-point function additionally needs the external operator basis and two-point normalization, the exchanged spectrum and multiplicities, long-multiplet dimensions, OPE coefficients, superconformal blocks in the same conventions, and a crossing-symmetric reflection-positive solution. Continue with the superconformal handoff.

Where to continue

• To realize the algebra on fields and auxiliary components, continue to component multiplets and closure and then superspace supertranslations.
• To see the same positivity structure become a Hamiltonian and a cohomology problem, continue to supercharges and partner Hamiltonians.
• To connect saturated central charges to solitons and vacuum geometry, continue to BPS particles and central charges.
• To apply the on-shell representation to scattering, continue to three-point amplitudes.
• To use superconformal multiplet data in crossing equations, continue to superconformal Ward identities and superconformal blocks and crossing.
• Return to the Supersymmetry and Duality volume for the broader path through superspace, dynamics, vacua, anomalies, localization, and duality.