Large-N limits

A large number of colors is only one part of a limit. Record the coupling, level, masses in radius units, observable normalization and operator size that stay fixed. The introductory large-N lesson derives the scaling rules; this page helps select the right rule for a formula already in hand.

Here $N$ is the fundamental color-space dimension. The ranks of $SU(N)$ and $U(N)$ are $N-1$ and $N$. Increasing $N$ changes neither spacetime dimension nor the number of supersymmetry generators.

Choose the family first

Theory and limit	What stays fixed	Expansion being organized
Four-dimensional $\mathcal N=4$ SYM, planar	$\lambda_{\rm YM}=g_{\rm YM}^2N$, global group choice, fixed-size observable	Color corrections, often in $1/N^2$ for the specified normalized single-trace observables
Equal-node ABJM, planar	$\lambda_{\rm AB}=N/k$, compact $U(N)_k\times U(N)_{-k}$, positive integer levels along the sequence	The genus/color expansion at fixed coupling ratio
The same ABJM theory, fixed level	A positive integer $k$ and the sphere-source prescription	Large rank with $\lambda_{\rm AB}\to\infty$; the sphere free energy begins at $N^{3/2}$

SYM has sixteen real Poincaré supercharges. The ABJM family has twelve at generic level, with enhancement to sixteen at $k=1,2$. The usual four-dimensional $\mathcal N=1$ and three-dimensional $\mathcal N=2$ chiral presentations make four manifest; the SYM circle’s four-dimensional $\mathcal N=2$ localization presentation makes eight manifest. Background and observable preservation are additional conditions, explained in the linked calculations. See multiplets and charges for these distinctions.

For rational $\lambda_{\rm AB}=p/q$, use integers $N=pn$, $k=qn$. An irrational target can be approached by integer ratios. Holding a finite integer level fixed while claiming to hold $N/k$ fixed is inconsistent as $N$ grows. AB, section 2.2, and FG, (1.3)–(1.4), identify the distinct limits.

Read the small parameter attached to the observable

Supplied result	Correct interpretation	Keep separate
$W_N/W_0=1+g(\lambda)/N^2+O_\lambda(N^{-4})$	A relative finite-color expansion at fixed $\lambda$	A strong-coupling expansion of $W_0$ or $g$
$W_0=2I_1(\sqrt\lambda)/\sqrt\lambda$	An exact planar function of positive coupling	The large-$\lambda$ approximation to the Bessel function
$F(N,k)\sim a_kN^{3/2}$, $a_k=\pi\sqrt{2k}/3$	Leading round-sphere free energy at fixed $k$, with $F=-\log Z$	A planar formula at fixed $N/k$, or a thermal free energy
$Z_{\rm pert}=e^{A(k)}C^{-1/3}\operatorname{Ai}[C^{-1/3}(N-B)]$	Resummation of the perturbative fixed-level sector	Exponentially small sectors omitted from the full $Z$

For the first two rows, $W$ is the normalized fundamental supersymmetric circle in the undeformed round-four-sphere localization background, and $I_\nu$ is the modified Bessel function. The ABJM rows use the undeformed, zero-mass supersymmetric round-three-sphere background and its fixed absolute normalization. These are the background and observable choices of the linked calculations. The finite-N lesson supplies the circle’s Gaussian definition and exact answer. At the same physical $\lambda$, removing the central $U(1)$ gives

$$W_N^{SU}=e^{-\lambda/(8N^2)}W_N^U, \qquad g_{SU}(\lambda)=g_U(\lambda)-\frac\lambda8. \tag{RF09.1}$$

The planar limits agree, but their first corrections differ. The source formulas are DG (2.26)–(2.27) and DG (5.1). The ordinary trace has an additional factor $N$ compared with the normalized trace. Fix the group and trace before comparing an error.

For the last row, $\operatorname{Ai}$ is the decaying Airy function,

$$C=\frac2{\pi^2k},\qquad B=\frac{k}{24}+\frac1{3k}. \tag{RF09.2}$$

The rank-independent normalization $A(k)$ also matters for absolute $Z$. FG (5.67) explicitly retains an additional term. The Airy lesson fixes the current constant-term input and the omitted-sector interpretation; its source guide records the old proposal that was withdrawn.

At fixed $k$, expanding the leading shifted power gives a relative correction $-3B/(2N)$ to $a_kN^{3/2}$. Logarithmic and constant terms in $F$ have relative size $O_k(\log N/N^{3/2})$. Thus the first relative correction here is of order $1/N$, even though the planar color expansion of a different family is organized in $1/N^2$. The comparison lesson derives this statement and tests it with rank differences.

Worked choice: follow the rank shift along two sequences

Take the same symbolic function $B(k)/N$ from (RF09.2). At fixed $k=4$ and $N=12n$ it is $1/(48n)$ and tends to zero. Along $N=12n$, $k=n$, the ratio $N/k=12$ stays fixed, but

$$\frac{B(n)}{12n}=\frac1{288}+\frac1{36n^2} \longrightarrow\frac1{288}. \tag{RF09.3}$$

The algebra is exact for these allowed integer sequences. It demonstrates why an expansion that treats $B/N$ as a vanishing fixed-level correction cannot simply be reused as a fixed-coupling rank expansion. At large fixed $\lambda_{\rm AB}$ its limiting contribution is instead $1/(24\lambda_{\rm AB})$, a strong-coupling correction.

A separate planar calculation is available: DMP (5.34)–(5.37) uses the shifted coupling and retains constant and exponential contributions. At leading strong coupling its sign-translated sphere result is $F/N^2\sim\pi\sqrt2/(3\sqrt{\lambda_{\rm AB}})$. That agreement has an independent source; substituting $k=N/\lambda$ into a fixed-$k$ leading term would not establish a uniform remainder estimate. The site’s $F=-\log Z$ is opposite to the free-energy sign in those source formulas.

Two checks before using a gravitational approximation

Large rank and small curvature are distinct requirements. For ABJM’s type-IIA description, the source’s conditions are $\lambda_{\rm AB}\gg1$ and $\lambda_{\rm AB}^{5/2}/N^2\ll1$. The first suppresses string-scale curvature; the second suppresses string loops. The eleven-dimensional regime instead includes $N\gg k^5$. These are parametric regimes, not numerical error tolerances. AB (4.5)–(4.7) and the holographic dictionary give their geometric meaning.

For example, the fixed-ratio sequence $N=12n,k=n$ makes the loop parameter $12^{5/2}/N^2$ decrease, while the curvature parameter $1/\sqrt{12}$ stays fixed. Increasing rank alone cannot make that curvature correction arbitrarily small. At fixed $k=4$, $N/k^5=N/1024$ instead grows. No particular finite $N$ becomes certified merely from the inequality symbol $\gg$.

Finally, distinguish the physical truncation from numerical evaluation. More digits can improve an integral or Bessel evaluation, but cannot restore a discarded genus, change the group, or supply an omitted exponential sector. A useful comparison names the observable, sequence and residual being tested; the Gaussian-loop lab provides a complete example.