Glossary

Start with the object a word describes. A particle in a massive vacuum, a local operator at a conformal fixed point, and a loop inserted in a path integral obey different tests. The worked comparison below makes these distinctions explicit. Each lookup entry links to the calculation that teaches it.

Fields and supersymmetry

Term	Meaning and first calculation
Supercharge	An odd symmetry generator that relates bosonic and fermionic objects. Its graded algebra includes translations. Start with what a supercharge does.
Multiplet	Fields or states related by the chosen symmetry algebra. Specify which of these is meant: an off-shell field list is not a particle-state count. Compare the multiplets.
Full, manifest, preserved supersymmetry	Full refers to the theory’s symmetry; manifest to what a presentation displays; preserved to the subset compatible with a background, state or insertion. The same $\mathcal N$ means different real-supercharge counts in different dimensions. See the counting table.
Chiral field or superfield	A constrained multiplet, written $\bar D\Phi=0$ in flat superspace, with scalar, fermion and auxiliary components. It may carry gauge charge. Chirality alone does not make its scalar a physical superconformal primary. See superspace and the component multiplet.
Auxiliary field; off shell; on shell	An auxiliary field has an algebraic equation of motion in the action under discussion. Off-shell identities hold without imposing field equations; on-shell identities use them. Auxiliary fields can complete the transformation algebra without adding propagating particle modes. See chiral closure.
R symmetry; flavor symmetry	An R symmetry acts on supercharges; a flavor symmetry commutes with them. A trial R symmetry may mix with Abelian flavor symmetries. Its charge need not be the superconformal charge that fixes scaling dimensions. See R and flavor symmetries.
Superpotential; Kähler potential	In a chiral-field description, the holomorphic superpotential determines F-term interactions; the Kähler potential determines kinetic geometry and associated D-term interactions. Neither word means the full scalar potential. See superpotential interactions.

The flat chiral constraint and its components are displayed in Bilal, section 4.2; curved three-dimensional components are given in P, section 6.2. Euclidean tilded fields are independent variables until an integration contour or reality condition is specified; see Euclidean continuation.

Vacua, sources, and geometry

Term	Meaning and first calculation
Vacuum; modulus; moduli space	A vacuum is a lowest-energy state. A modulus continuously labels inequivalent vacua; the moduli space includes the appropriate gauge identifications. A classical flat direction need not survive quantum effects. See Higgs and Coulomb branches.
Coulomb branch; Higgs branch	Names for branches characterized, respectively, by vector-multiplet scalar data and matter expectation values. Determine the unbroken gauge group rather than inferring it from the name. See the SQED example.
Central charge; BPS particle	A central charge commutes with the super-Poincaré generators. A BPS particle saturates the associated mass bound and belongs to a shorter massive representation. A charge row and a bound do not prove that a stable particle exists. See the positive algebra and particle stability.
Current; source; current multiplet	A current is an operator; a source is an externally specified field coupled to an operator. A current multiplet collects related operators, including conservation laws with their equations-of-motion and contact-term qualifications. See source variation and the R-multiplet.
Rigid supersymmetric background	Fixed metric and background fields admitting the required spinor parameters. Their Killing-spinor equations test the preserved symmetry; the fixed fields need not solve dynamical supergravity equations. See the background equations.
Spinor section; global solution	A spinor has components in local frames and prescribed changes on overlaps. A local solution of a differential equation need not obey the global patch or periodicity conditions. See spinor bundles.
Wilsonian effective action; 1PI action	A Wilsonian action results from integrating out modes above a chosen scale while retaining specified light fields. The one-particle-irreducible action generates proper vertices and can contain infrared nonlocality. Their holomorphy statements are not interchangeable. See nonrenormalization and its limits.

The current/source distinction is explicit in P, (2.10). The effective-action distinction is explained in Intriligator–Seiberg, section 2.3.

Operators and observables

Term	Meaning and first calculation
Primary; descendant	In radial quantization a conformal primary is annihilated by special conformal generators; translations create descendants. A superconformal primary is also annihilated by conformal supercharges. See states and local operators.
Shortened operator multiplet	A superconformal representation with null descendants removed from its physical state space. Shortening need not truncate the ordinary derivative tower: the free scalar below still has infinitely many descendants. See shortening and positivity.
Chiral operator; chiral ring	A gauge-invariant local operator annihilated by the specified barred supercharges is chiral. Taking the appropriate closed operators modulo exact ones defines the chiral-ring classes. Their relations need not equal a naive classical F-term quotient. See chiral rings and vacua.
Q-closed; Q-exact; cohomology	Closed means $Q\mathcal O=0$; exact means $\mathcal O=Q\mathcal V$. Cohomology identifies closed objects differing by exact ones. If $Q^2$ is a bosonic symmetry rather than zero, first specify the invariant subspace on which this construction applies. See localization.
Protected	A specified quantity is constrained under specified deformations. State whether this means a scaling dimension, an index, or a correlation function, and which hypotheses prevent recombination or boundary/contact contributions. See what protection does and does not imply.
Supersymmetric observable; index	A supersymmetric insertion is invariant under a chosen charge. An index is a graded trace in which paired states cancel, with charges, fugacities and convergence specified. Neither is automatically a count of all states or a coupling-independent number. See the index definition.
Wilson loop; monopole operator	A Wilson loop transports gauge charge around a curve; supersymmetry generally requires extra scalar or matter couplings. A three-dimensional local monopole operator specifies magnetic flux through a small surrounding sphere. It is not the smooth four-dimensional monopole particle. See supersymmetric loops and monopole operators.
Duality	An equivalence between stated physical sectors or regimes, with a dictionary for operators, charges and parameters. An infrared dual gauge field need not be an additional microscopic field of the original theory. See Seiberg duality.

For radial adjoints and null states, see Simmons-Duffin, sections 7.2–7.3. The selected three-dimensional index algebra appears in Kapustin–Willett, section 2.1.

Worked disambiguation: five objects, two different algebras

A massive particle. Work in a Lorentzian four-dimensional N=2 Coulomb vacuum preserving eight real supercharges. Write the central charge entering the algebra as $\mathcal Z$. In this site’s first Seiberg–Witten period units,

$$\mathcal Z=\sqrt2(n_m a_D+n_ea),\qquad M\geq|\mathcal Z|.$$

The periods and $\mathcal Z$ have mass dimension one; $n_m,n_e$ are dimensionless particle-charge labels, with the W boson assigned $n_e=1$. Bilal’s massive-algebra central entry is $Z_{\rm BIL}=2\mathcal Z$. These factors must accompany a comparison of bounds; the extended-algebra lesson derives them.

As an algebraic test, take $\mathcal Z=(5+12i)\mu$, where $\mu>0$ is a mass unit. One positive anticommutator block is

$$G=2\begin{pmatrix}M&\mathcal Z\\ \mathcal Z^*&M\end{pmatrix}, \qquad \operatorname{eig}G=2(M\pm13\mu). \tag{RF05.1}$$

At $M=13\mu$ a null eigenvector is $(-(5+12i)/13,1)$. Positivity makes both the associated charge combination and its adjoint annihilate the state. The other spinor block gives the corresponding half-BPS shortening. At $M=14\mu$ both eigenvalues are positive. This tests a representation conditional on the state’s existence; it neither locates a vacuum realizing these illustrative period data nor establishes its particle spectrum. The source algebra is Bilal, (3.12)–(3.16).

A shortened local operator. Now change theories and vacua: take a free, gauge-neutral chiral multiplet in a three-dimensional N=2 superconformal vacuum. There are four real Poincaré and four conformal supercharges. Its scalar has

$$\langle\phi(x)\bar\phi(0)\rangle=\frac1{4\pi|x|}, \qquad \Delta_\phi=r_\phi=\frac12.$$

Radial conjugation pairs a selected Poincaré charge $q$ with a conformal charge, and $\{q,q^\dagger\}=D-R-J_3$. On the scalar primary, $q^\dagger|\phi\rangle=0$ and $j_3=0$, so

$$\|q|\phi\rangle\|^2=(\Delta_\phi-r_\phi)\|\phi\|^2=0. \tag{RF05.2}$$

Here $1/2$ is a dimensionless scaling dimension and R-charge, not a particle mass or the central charge of (RF05.1). The radial calculation also checks the free scalar’s null equation-of-motion descendant. Substituting an arbitrary trial R-charge into (RF05.2) would change the algebra unless the compensating flavor term were retained.

The chiral field. The superfield containing this $\phi$ satisfies a chirality constraint. Adding a supersymmetric mass preserves that constraint but introduces a scale: the preceding conformal-vacuum shortening argument no longer applies. For gauge-charged elementary fields there is an additional distinction: they are not themselves gauge-invariant physical local operators.

The current multiplet. In the three-dimensional description, $j_R^\mu$, the supersymmetry currents and $T_{\mu\nu}$ are operators in the R-multiplet. In P’s source convention the linear action contains $j_R^\mu(A_\mu-\tfrac32V_\mu)$. The fixed fields $A,V$ are sources, not further conserved currents; additional operators complete the multiplet. Conservation holds with the usual field-equation or separated-insertion qualifications. The R-multiplet calculation exhibits the full coupling.

The supersymmetric observable. For an insertion $\mathcal O$ satisfying $Q\mathcal O=0$, a normalized Euclidean expectation value with weight $e^{-S(t)}$ still obeys

$$\partial_t\langle\mathcal O\rangle =\langle\partial_t\mathcal O\rangle -\langle\mathcal O\,\partial_tS\rangle_c. \tag{RF05.3}$$

The subscript $c$ subtracts the product of expectations. Differentiating the numerator and denominator derives the two terms. Independence follows only if the insertion is fixed and the action variation is Q-exact under valid Ward identities, including measure, contour and boundary/contact conditions. Being Q-closed by itself proves neither (RF05.1) nor parameter independence. Field-theory localization supplies the full argument.

Check the distinction

1. Keep $\mathcal Z=(5+12i)\mu$ but propose $M=12\mu$. Is this a non-BPS massive state?
2. Replace the free scalar’s exact R-charge by $r_t=1/2+t$ using a flavor symmetry with $F\phi=\phi$. Does $t>0$ disprove unitarity?
3. Change a background source coupled to a current. Does conservation force the generating functional to be independent of that source?

Answers

1. No. The smaller eigenvalue of (RF05.1) is $-2\mu$, forbidden by positivity. A non-BPS state in this charge sector must instead lie above the bound, if it exists.
2. No. Writing $R_t=R+tF$ requires $D-R-J_3=D-R_t-J_3+tF$. Its value on $\phi$ remains zero. The exact superconformal R-charge is the one appearing without an extra flavor term.
3. No. Conservation constrains source dependence through Ward identities; it does not eliminate physical responses. A second source derivative contains a connected current correlator and, where applicable, a contact term. Follow the signs in sources and generating functionals.