Theory dictionary and methods

“Find an exact result” is not yet a calculation. A vacuum equation, a protected index and a sphere partition function answer different questions, even when they concern the same theory. This dictionary identifies the theory first and then selects a useful first step. For component counts and auxiliary fields, use the separate multiplet reference.

Read the labels separately

Full supersymmetry is a property of the interacting action. Manifest supersymmetry is the subalgebra made explicit by a chosen notation. Changing notation does not remove the other charges. Background-preserved supersymmetry is determined by globally defined solutions of the appropriate rigid spinor equations, including the source fields and boundary conditions. A chosen vacuum can break symmetries that both the action and background admit.

The tables below count real Poincaré supercharges of the flat Lorentzian action. Conformal supercharges are additional generators. On a Euclidean curved space, independent complex spinor parameters require their own reality and global prescription; their component count must not be read as an automatic doubling of the Lorentzian theory. The background-field reference explains which data must be specified before counting preserved charges.

Gauge rank is another number: $\operatorname{rank}SU(N)=N-1$, whereas $\operatorname{rank}U(N)=N$. The color parameter $N$ does not count supercharges.

Recurring theories

In the “full / manifest” column, the numbers are real supercharges. A chiral contains a complex scalar and its spinor partner; a vector contains a gauge connection and its supersymmetry partners. A superpotential $W$ specifies holomorphic interactions. These ingredients still need the corresponding kinetic terms and supersymmetric completion.

Theory and dimension	Full / manifest in the usual notation here	Gauge group and rank	Defining fields and interactions
Supersymmetric quantum mechanics, one dimension	$2/2$: one complex $Q$ and its adjoint	None in the oscillator example	A coordinate, a two-state fermionic space, and a specified supercharge with a Hilbert-space domain
Free chiral or XYZ, three dimensions	$\mathcal N=2$: $4/4$	None	One chiral with $W=0$, or three chirals $X,Y,Z$ with $W=\lambda XYZ$, $\lambda\ne0$; these are different models
One-flavor SQED, three dimensions	$\mathcal N=2$: $4/4$	Compact $U(1)$, rank one	A Yang–Mills vector and two chirals of charges $+1,-1$; initially gauge Chern–Simons level, $W$, masses and FI parameter are zero
Wess–Zumino matter, four dimensions	$\mathcal N=1$: $4/4$	None	Chirals with specified Kähler kinetic metric and $W$; supersymmetric action does not guarantee a supersymmetric vacuum
SQCD, four dimensions	$\mathcal N=1$: $4/4$	$SU(N_c)$, rank $N_c-1$	A vector and $N_f$ fundamental/anti-fundamental chiral pairs; flavor number, masses and energy regime matter
Pure Seiberg–Witten theory, four dimensions	$\mathcal N=2$: $8/4$ in $\mathcal N=1$ notation	$SU(2)$, rank one	One $\mathcal N=2$ vector, equivalently an $\mathcal N=1$ vector plus an adjoint chiral, with the required relative couplings; no hypermultiplet
Four-dimensional $\mathcal N=2$ theory with fundamental matter	$8/4$ in $\mathcal N=1$ notation	$SU(2)$, rank one in the matter examples	An $\mathcal N=2$ vector and $N_f$ full fundamental hypermultiplets with specified masses and vector–matter couplings; this is not the four-supercharge SQCD row
$\mathcal N=2^*$, four dimensions	$8/4$ at nonzero physical hypermultiplet mass; $16/4$ at zero mass	$SU(2)$, rank one in this example	An $\mathcal N=2$ vector plus one full adjoint hypermultiplet; its mass deforms the $\mathcal N=4$ action
$\mathcal N=4$ Yang–Mills, four dimensions	$16/4$ in $\mathcal N=1$ notation, or $16/8$ in $\mathcal N=2$ notation	Usually $SU(N)$, rank $N-1$	A gauge field, six real adjoint scalars and four independent Weyl fermions; commutator potential and Yukawa couplings are fixed together
ABJM, three dimensions	Generic $\mathcal N=6$: $12/4$ in $\mathcal N=2$ notation	$U(N)_k\times U(N)_{-k}$; rank $N$ per factor, $2N$ in total	Two Chern–Simons vectors; $A_{1,2}$ in $(\mathbf N,\overline{\mathbf N})$, $B_{1,2}$ in the reverse representation; the specified quartic interaction is essential

The SQED baseline and its Coulomb variables are described in Aharony et al., §§2–3; the four-dimensional SQCD family in Intriligator–Seiberg, §4.1. For extended theories, compare Minahan, §2 and ABJM, §§2.1–2.2.

Three qualifications prevent common misidentifications. Adding three arbitrary adjoint chirals to a vector does not establish $\mathcal N=4$: the interactions must fit. Replacing $SU(N)$ by $U(N)$ changes global data; its locally free central multiplet does not justify silently changing the group in an observable. In standard ABJM at $|k|=1,2$, monopole operators supply the enhanced $\mathcal N=8$ current multiplets; scalar-potential invariance alone is not that argument. A pure Chern–Simons vector has no independent Maxwell photon polarization.

The two meanings of N=2

The same symbol hides a factor of two in supercharges and a substantial difference in dynamics.

Question	Three-dimensional $\mathcal N=2$	Four-dimensional $\mathcal N=2$
How many real Poincaré charges?	Four	Eight
What is the Yang–Mills coupling dimension?	$[g_3^2]=1$	$[g_4^2]=0$
Which gauge terms are relevant to the comparison?	Yang–Mills and supersymmetric Chern–Simons terms; allowed levels depend on global group, matter and parity-anomaly prescription	Yang–Mills and a theta term $\int\operatorname{Tr}F\wedge F$; the latter is not a three-dimensional Chern–Simons action
What describes a generic unlifted Abelian Coulomb patch?	A real vector scalar $\sigma$ and a periodic scalar dual to the photon, for each Abelian factor	A complex vector scalar $a$ and an Abelian photon, for each factor
What organizes useful exact results?	Holomorphic constraints, monopole operators, and specified sphere or index localization	Local prepotentials, periods and central charges; specified localization calculations provide other observables

The coupling dimensions follow without a SUSY assumption. Absorb the coupling into the connection, so $D_\mu=\partial_\mu-i a_\mu$, $[a_\mu]=1$ and $[F_{\mu\nu}]=2$. In the dimensionless action $g_d^{-2}\int d^dx\,\operatorname{Tr}F^2$, the integrand has net mass dimension $4-d-[g_d^2]$. Therefore

$$[g_d^2]=4-d. \tag{RF08.1}$$

A dimensionless four-dimensional coupling can still run; a dimensionful three-dimensional coupling does not by itself determine the infrared theory. The dual photon is a replacement for the three-dimensional Abelian photon, not an additional propagating mode. Its period and its normalization relative to $\sigma$ require a charge convention. Quantum effects can change or lift a $3d\ \mathcal N=2$ Coulomb branch; Aharony et al., §6.1, (6.2) give a monopole-generated superpotential example.

Ordinary spatial reduction preserving all charges instead gives

$$4d\ \mathcal N=2\ \longrightarrow\ 3d\ \mathcal N=4, \qquad 8\longrightarrow8. \tag{RF08.2}$$

On a regular rank-$r$ Coulomb patch, the two real components of each $a$, a circle holonomy and a dual photon give $4r$ real scalar coordinates. For a circle of circumference $L$, integration of a coordinate-independent kinetic term gives $g_3^2=g_4^2/L$ in the same connection normalization. This is a low-energy starting point, not the entire finite-radius quantum metric. Seiberg–Witten, introduction and §2.1, distinguish classical reduction from quantum compactification; the reduction bridge fixes the circle, periods and charge units explicitly.

Similarly, the star in $\mathcal N=2^*$ denotes a particular mass deformation, not an extra supercharge. The mass is assigned to the adjoint hypermultiplet. A mass for the chiral inside the vector is a different deformation and generally leaves only $\mathcal N=1$. The source construction is Seiberg–Witten, §16.2. Its adjoint-theory periods use the first-paper normalization; neither the fundamental-matter charge basis nor a localization equivariant mass may be substituted without the explicit dictionary.

From a question to a first calculation

Question	Start with	What must still be checked
Does this configuration define a SUSY vacuum?	Solve auxiliary $F$- and $D$-equations and quotient by gauge equivalence; see vacuum tests	Kinetic metric, stability, boundary/runaway behavior, and quantum changes to the effective theory
Is this particle BPS or this operator protected?	Identify the relevant algebra, charges and shortening condition; see BPS and protected quantities	Existence and stability of a particle, or possible recombination of an operator multiplet
Which states does an index retain?	Specify the Hilbert space, $Q$, its positive anticommutator and commuting fugacities; see the index definition	Bose–Fermi cancellations, convergence, boundary conditions and accidental symmetries
What is a curved-space partition function?	Fix geometry, rigid sources, charge assignments, contour and local counterterms; then choose a valid localization deformation	Saddle measure, zero modes, determinant prescription and the precise observable
Are two descriptions dual?	Write operator, charge, parameter and deformation maps; compute one common quantity in a stated regime	Contact phases, global data and the scope of the proposed equivalence; agreement of one test is not a proof

For a small vacuum diagnostic, take flat $4d\ \mathcal N=1$ gauge-free matter $X$, Kähler potential $K=Z\bar X X$ with constant $Z>0$, and $W=fX$. Here $[X]=1$, $[f]=2$, and $Z$ is dimensionless. In the site’s auxiliary convention,

$$\mathcal L_{\rm aux}=Z\bar F F+fF+\bar f\bar F, \qquad F=-\frac{\bar f}{Z}, \qquad V=\frac{|f|^2}{Z}. \tag{RF08.3}$$

For $f\ne0$, no constant scalar value makes $F$ vanish. This particular model has a flat scalar potential and no isolated minimum. Holomorphic data decide the absence of an $F=0$ point here, but $W$ alone does not give the energy: the kinetic normalization enters. The effective-action lesson explains that distinction beyond this constant-metric example.

For a duality diagnostic, take electric $4d\ \mathcal N=1$ $SU(3)$ SQCD with five massless flavors. The proposed magnetic gauge group is $SU(5-3)=SU(2)$, with five pairs and $25$ meson chirals; its rank is one, versus electric rank two. Comparing ultraviolet gauge-boson counts would test the wrong claim. The worked anomaly and deformation check tests a shared infrared description. This is the magnetic construction of Intriligator–Seiberg, §5.3, (5.5).

For a partition-function diagnostic, “ABJM at finite $N,k$ on the round sphere” points first to the specified matrix integral and small-rank checks. A large-$N$ formula needs a separately justified limit. A circular Wilson loop in four-dimensional $\mathcal N=4$ instead requires an operator insertion and its own finite-$N$ normalization. Equal amounts of supersymmetry would not make those two observables interchangeable.

Protection and duality have a scope

Holomorphy constrains a Wilsonian superpotential; it does not generally fix a Kähler metric or forbid nonperturbative superpotentials. A Wilsonian action integrates out modes above a scale, whereas a full one-particle-irreducible action also includes infrared propagation. These distinctions are explicit in Intriligator–Seiberg, §§2.1–2.4.

Extended supersymmetry can constrain more: the local prepotential controls the two-derivative Abelian $4d\ \mathcal N=2$ action. That result does not determine every momentum-dependent amplitude. Likewise, half-BPS operators and Konishi illustrate why a supersymmetric theory contains both protected and unprotected dimensions. A central-charge formula fixes the mass if the corresponding BPS particle exists and is stable; the formula alone is not a list of the spectrum.

Duality language	Intended comparison	A necessary distinction
Seiberg–Witten electric/magnetic descriptions	Local low-energy descriptions patched over a Coulomb branch	A change of period and charge basis must preserve the central charge; this is not equality of arbitrary ultraviolet Lagrangians
Seiberg duality	Different gauge theories flowing to the same infrared physics in a specified regime	Gauge ranks and elementary fields can differ
$\mathcal N=4$ electric/magnetic duality	Couplings, electric/magnetic charges and complete theory data transform	Global gauge groups and line operators determine which theories are exchanged
SQED–XYZ duality	The selected three-dimensional $\mathcal N=2$ infrared pair, including monopole/meson and deformation maps	This four-supercharge example is not automatically the eight-supercharge $\mathcal N=4$ mirror pair
Field/gravity duality	A boundary quantum field theory and a gravitational/string description	A classical gravity calculation needs its own large-rank and curvature/coupling regime; see the ABJM dictionary

Three quick decisions

1. Change the kinetic normalization

In the gauge-free example, replace $Z$ by $4Z$ while holding the written $f$ fixed. Does this restore supersymmetry? What changes after canonical normalization?

Solution

The auxiliary becomes $F=-\bar f/(4Z)$, still nonzero, and $V=|f|^2/(4Z)$. With $X_c=\sqrt{4Z}\,X$, the canonical superpotential coefficient is $f_c=f/\sqrt{4Z}$, so $V=|f_c|^2$. This changes the canonically normalized coupling, not the existence of an $F=0$ solution. If instead $f_c$ were held fixed, the energy would stay fixed. This is why the phrase “hold the coupling fixed” needs a normalization.

2. Reduce a rank-one theory

Compactify pure four-dimensional $\mathcal N=2$ $SU(2)$ theory on a spatial circle with periodic fermions, no R-twist, and a regular Coulomb vacuum. Is the generic low-energy result three-dimensional $\mathcal N=2$? How many scalar coordinates remain after dualizing the photon?

Solution

All eight real charges survive this untwisted construction, so the three-dimensional label is $\mathcal N=4$. Rank one gives two real components of $a$, one circle holonomy and one dual photon: four real scalar coordinates. The photon is replaced, so it is not a fifth coordinate. The local count does not fix the global period lattice or the quantum metric; those require the group, charge units and finite-radius dynamics. An antiperiodic thermal circle would invalidate the supersymmetry-preservation assumption.

3. Interpret matching sphere integrals

Two candidate dual descriptions have equal absolute values of their sphere partition functions for several masses. Is the full partition-function identity established? Does it follow that their unprotected operator dimensions agree term by term in ultraviolet perturbation theory?

Solution

Neither conclusion follows. First compare the full complex functions in matched source and local-counterterm conventions; contact terms can affect their phases, and finitely many masses do not establish an identity. Even an established sphere identity is one protected test. An infrared duality compares infrared observables under the operator map, not term-by-term ultraviolet expansions of different elementary fields. The next useful step is a source-sensitive phase or deformation test, rather than a stronger claim from the same data.