Notation

The same letter can denote a field, a charge or an observable. Read its definition, indices and theory before combining formulas. This page provides a lookup guide and one exact source translation. It does not impose one notation on every paper.

Dimension, supersymmetry and gauge size

Label	Meaning	What it does not determine
$d$	Spacetime dimension; the signature is stated separately	Spinor reality or a supersymmetry count
$\mathcal N$	Copies of the minimal spinor supercharge in a specified dimension	Gauge rank
Full supersymmetry	The complete Poincaré algebra of the flat theory	How many charges a chosen presentation makes explicit
Manifest supersymmetry	The subalgebra explicit in the chosen variables	A smaller physical theory
Background-preserved supersymmetry	Charges whose transformations preserve the specified metric and all other sources	A number inferable from the metric alone
$N$ in $SU(N)$ or $U(N)$	Dimension of the fundamental color space	The number of generators or supercharges

In three dimensions, $\mathcal N=2$ has four real Lorentzian Poincaré supercharges; in four dimensions, $\mathcal N=2$ has eight. Four-dimensional $\mathcal N=4$ SYM has sixteen, even when written in $\mathcal N=1$ variables that make four manifest. Standard compact $U(N)_k\times U(N)_{-k}$ ABJM has twelve at generic level, with four manifest in its $\mathcal N=2$ chiral presentation. Its exceptional levels $k=1,2$ have sixteen; the ABJM introduction explains the global theory and the enhancement result.

These counts exclude the additional conformal supercharges of a superconformal algebra. In the quantum-mechanics examples, one complex charge and its adjoint supply two real generators. Independent Euclidean spinor variables do not double the count of the Lorentzian parent. See multiplets and charges for the detailed dimension-dependent tables.

Strictly, $\operatorname{rank}SU(N)=N-1$ and $\operatorname{rank}U(N)=N$, while their Lie-algebra dimensions are $N^2-1$ and $N^2$. The phrase “large rank” often uses $N$ as the growing size parameter. In equal-size ABJM, each node has rank $N$ and the product group has rank $2N$. A Lie algebra alone does not specify the global gauge group, allowed fluxes or line operators.

Geometry, charges and fields

Symbol	Usual role here	Essential qualification
$\ell$	Physical sphere radius	$[\ell]=-1$ in mass units; a dimensionless mass is $\ell m$
$r$	R-charge of a chiral scalar	Some geometric coordinates also use $r$ locally; check the definition
$R$	Ricci scalar, an R-symmetry generator, or a source paper’s radius	These are different objects; a Ricci scalar has dimension two
$\mu,\nu$; $a,b$; $\alpha,\beta$	Often coordinate, orthonormal-frame and spinor indices	The same letters can label other objects when explicitly defined
$A_\mu$, $a_\mu$	Connections	In the three-manifold package $A_\mu$ is a fixed R-source; $a_\mu$ commonly denotes a dynamical gauge connection
$a$, $a_D$	Electric and magnetic special coordinates in Seiberg–Witten theory	They are scalar periods, not connection one-forms
$D_\mu$, $D$	Covariant derivative, vector-multiplet auxiliary	An index distinguishes these uses; neither is a spacetime dimension
$\lambda_\alpha$	Gaugino	The unindexed $\lambda$ often denotes a coupling instead
Hat, tilde, bar, dagger	A second node, an independent partner, or a conjugation operation	A hat on an ABJM field is not complex conjugation; Euclidean tilded fields need not be conjugates

The original three-manifold paper uses $R=-6/\ell^2$ on the round sphere. The spinor and curvature reference fixes the matrices, index order and curvature trace. The background-field reference distinguishes prescribed sources from dynamical auxiliaries and the matter connection $A-V/2$ from the current source $A-3V/2$. Changing a symbol does not make those two connections interchangeable.

Several different quantities called F or W

Notation	Definition or role	Where to continue
$F_\Phi$	Chiral auxiliary component; a source may write simply $F$	Chiral multiplet
$F_{\mu\nu}$	Gauge-field strength	Its connection and sign convention are specified with the action
$F=-\log Z$	Free energy from a partition function	$\operatorname{Re}F=-\log\lvert Z\rvert$; phases and counterterms still matter
$\mathcal F(a)$	Holomorphic prepotential in four-dimensional $\mathcal N=2$ theory	Low-energy prepotential
$W(\Phi)$	Superpotential, a holomorphic interaction	Superpotential interactions
$W[C]$ or $W_{\rm loop}$	Wilson-loop operator on a contour $C$	Supersymmetric Wilson loops

For the prepotential, $a_D=\partial_a\mathcal F$ and $\tau_{\rm eff}=\partial_a^2\mathcal F=da_D/da$ in the specified local electric frame. In general this is not $a_D/a$. The source writes the holomorphic function as $F$; calling it $\mathcal F$ here changes no normalization. These definitions are directly in Seiberg–Witten v1, (2.10) and the paragraph before (3.2). The period and charge basis must still be fixed before comparing a mass formula. The paper guide explains the source versions and the published figure correction.

Couplings and overloaded local symbols

Symbol and setting	Meaning	Do not silently identify it with
$\lambda_{\rm SYM}=g_{\rm YM}^2N$	Four-dimensional ‘t Hooft coupling in the declared trace convention	ABJM’s $\lambda=N/k$
$\lambda_{\rm ABJM}=N/k$, $k>0$	Rank-to-level ratio for the ordered pair of opposite levels	A continuously variable level at fixed finite rank
$\tau_{\rm UV}=\theta/(2\pi)+4\pi i/g^2$	A specified four-dimensional microscopic complex coupling	A low-energy period ratio without a normalization map
$\tau_{\rm eff}=da_D/da$	Local low-energy coupling	$a_D/a$, or every other Abelian coupling convention
$\tau_{RR}$	Coefficient of the R-current two-point function	A gauge coupling or a coordinate
$\tau$ in a geometry or dual-photon calculation	A coordinate or a periodic scalar, as locally declared	The complex coupling above
$\tau_i$	Pauli matrices in some spinor formulas	Any of these scalar quantities
$q$	May be a charge eigenvalue, an instanton parameter, an index fugacity, or an odd supercharge	A common numerical value across those uses
$k$	Chern–Simons level; elsewhere an instanton number or mode label	The same lattice or orientation in every calculation

Even an expression such as $q=e^{2\pi i\tau}$ needs its definition: some instanton-counting formulas instead absorb powers of a dimensional scale into $q$. State its dimension and the quantity conjugate to it. Likewise, “large $N$” needs a sequence of theories: fixed $k$ and fixed $N/k$ are different ABJM limits. The canonical large-rank lesson defines those limits; the two coupling conventions above are used explicitly in Drukker–Gross, below (1.1) and ABJM, section 2.2.

Translate one expression while keeping the operator fixed

Drukker–Gross v2 equation (1.4) defines the normalized fundamental probe with an overall $i$:

$$W_{\rm DG}=\frac1N\operatorname{Tr}_N \mathcal P\exp\left[i\oint \bigl(A_\mu\dot x^\mu+i\Phi_i\lvert\dot x\rvert\theta_{\rm DG}^i\bigr)dt\right]. \tag{RF01.1}$$

Here $A_\mu,\Phi_i$ are Hermitian matrix fields with the coupling absorbed, and $\theta_{\rm DG}$ is a unit vector in scalar-flavor space. It is not the four-dimensional theta angle. Rename the connection $\mathcal A_\mu=A_\mu^{\rm DG}$, keeping the scalar matrices fixed. The site’s Wilson-loop operator can then be written as

$$W[n]=\operatorname{tr}_N\mathcal P\exp\left[\oint \bigl(i\mathcal A_\mu\dot x^\mu+n^i\Phi_i\lvert\dot x\rvert\bigr)dt\right], \qquad \operatorname{tr}_N:=\frac1N\operatorname{Tr}_N,\qquad n=-\theta_{\rm DG}. \tag{RF01.2}$$

Distributing the first $i$ gives $iA_\mu\dot x^\mu-\Phi_i\theta_{\rm DG}^i\lvert\dot x\rvert$. This proves equality point by point. The path, orientation, representation, gauge coupling and order of multiplication remain fixed. In particular $\operatorname{tr}_N\mathbf1_N=1$, whereas $\operatorname{Tr}_N\mathbf1_N=N$. Replacing the latter by the former inside an action also requires translating its coefficient.

One finite $U(2)$ matrix check makes the ordering visible. Let three consecutive constant segment generators be $K_1=E_{12}$, $K_2=E_{21}$ and $K_3=\operatorname{diag}(\log4,0)$, where $E_{ij}$ has a single unit entry. These are dimensionless integrated generators. Each can be written as $K_j=iA_j-S_j$ with Hermitian $A_j=(K_j-K_j^\dagger)/(2i)$ and $S_j=-(K_j+K_j^\dagger)/2$, so the source-to-site scalar-sign translation applies. With later segments multiplying on the left,

$$U=e^{K_3}e^{K_2}e^{K_1} =\begin{pmatrix}4&4\\1&2\end{pmatrix},\qquad \frac12\operatorname{Tr}U=3. \tag{RF01.3}$$

Both notations give this matrix. Reversing the product instead gives normalized trace $9/2$; keeping the wrong scalar sign gives $3/4$. Thus a correct translation preserves more than the names of the letters. This is a finite matrix test of the operator dictionary, not a claim that these arbitrary segment fields form a supersymmetric background or compute a Wilson-loop expectation value.