Large-N Limits, Normalizations, and Orders of Limits

A nontrivial large-$N$ limit is defined by an index family, a normalization of the action and fields, the couplings and operator normalizations held fixed, a state, and an order of limits. There is no universal instruction to “send $N$ to infinity.” Vector, matrix, gauge, and tensor theories keep different interactions finite, select different graph families, and can fail for different infrared or critical reasons.

Required background. What does nonperturbative mean? supplies the observable, regulator, and control data that must accompany a limiting claim. Running couplings and dimensional transmutation supplies the distinction between a bare $N$-scaling and a renormalized coupling held fixed at a declared scale.

Helpful background. The O(N) model as a strong-coupling laboratory supplies a physical vector-model application.

The data that define a large-N limit

Throughout this chapter, an index written $i=1,\ldots,N$ is a vector index, a pair $M^i{}_j$ is an adjoint or matrix index, and $T_{abc}$ is a rank-three tensor with each of $a,b,c=1,\ldots,N$. Unless a page says otherwise:

• the regulator, spacetime manifold, boundary conditions, state, and renormalization scale $\mu$ are fixed before the $N$ limit;
• dimensionless renormalized couplings named below are held fixed at $\mu$;
• invariant operators are normalized so their one-point functions remain $O(1)$;
• connected correlation functions are taken in one selected phase or clustering state;
• volume, continuum, critical, and infrared limits are written in their intended order.

These choices are part of the mathematical statement. Changing any one of them can change the power of $N$ or the leading saddle.

Four representative actions make the distinction concrete.

For an $O(N)$ vector field,

$$S_{\mathrm v} = \int\mathrm d^d x \left[ \frac12(\partial_\mu\phi_i)^2 +\frac{m^2}{2}\phi_i\phi_i +\frac{\lambda}{4N}(\phi_i\phi_i)^2 \right],$$

with $\lambda$ held fixed. Components $\phi_i$ have $O(1)$ propagators, a quartic vertex contributes $N^{-1}$, and every closed vector-index loop contributes $N$.

For a Hermitian matrix field,

$$S_{\mathrm m} = N\int\mathrm d^d x\, \operatorname{tr} \left[ \frac12(\partial_\mu M)^2 +\frac{m^2}{2}M^2 +\sum_{k\ge3}\frac{g_k}{k}M^k \right],$$

with every $g_k$ fixed. The Gaussian contraction is

$$\langle M^i{}_j M^k{}_l\rangle_0 \propto \frac1N\delta^i{}_l\delta^k{}_j.$$

Thus a ribbon propagator contributes $N^{-1}$, a single-trace vertex contributes $N$, and every closed index face contributes $N$.

For $SU(N)$ Yang–Mills, choose $\operatorname{tr}(T^aT^b)=\delta^{ab}/2$ and put the coupling outside the field strength:

$$S_{\mathrm{YM}} = \frac{N}{2\lambda_{\mathrm t}} \int\mathrm d^d x\, \operatorname{tr}F_{\mu\nu}F_{\mu\nu}, \qquad F_{\mu\nu} = \partial_\mu A_\nu-\partial_\nu A_\mu+i[A_\mu,A_\nu].$$

The ’t Hooft coupling $\lambda_{\mathrm t}=g_{\mathrm{YM}}^2N$ is held fixed. In this convention adjoint propagators and vertices have the same $N^{-1}$ and $N$ weights as the matrix theory. With $N_f$ fundamental flavors, the standard ’t Hooft limit also holds $N_f$ fixed; a fundamental loop makes a boundary and is suppressed by $N_f/N$ relative to a closed adjoint surface. The Veneziano limit instead keeps $N_f/N$ fixed and is a different expansion.

For the tensor example used here, take a real $O(N)^3$ field $T_{abc}$ with a Gaussian propagator of order one and the tetrahedral interaction

$$S_{\mathrm{int}} = \frac{g}{4N^{3/2}} \int\mathrm d^d x\, T_{a_1a_2a_3} T_{a_1b_2b_3} T_{b_1a_2b_3} T_{b_1b_2a_3},$$

with $g$ fixed. A vertex contributes $N^{-3/2}$ and each single-color closed strand contributes $N$. This is one model-specific tensor scaling, not a universal rule for all tensor invariants.

Large-N counting and topology map

The figure organizes the counts that follow from these normalizations. Inspect the two one-vertex ribbon contractions: the planar and crossed pairings have the same $V=1$ and $E=2$, but different face counts. Also inspect the adversarial row, where deleting the overall $N$ from the matrix action changes both graph powers and destroys the Euler-characteristic organization.

Counting map for the declared vector, Hermitian-matrix, adjoint-gauge, and rank-three $O(N)^3$ tensor normalizations. The dashed adversarial box changes the action normalization and therefore the answer. The frozen ribbon examples are exact combinatorial counts, while the family comparison is schematic and not a claim of a string or holographic dual.

For a connected vacuum ribbon graph with $V$ single-trace vertices, $E$ propagators, and $F$ closed index faces,

$$\mathcal A_{\mathcal G} \propto N^{V-E+F}.$$

Thickening the graph produces an orientable closed surface, so

$$V-E+F = \chi = 2-2h,$$

where $h$ is the genus. If $b$ unnormalized fundamental or source boundaries are present,

$$\mathcal A_{h,b} \propto N^{2-2h-b}.$$

An explicit factor $N^{-1}$ in each normalized trace operator supplies a further $N^{-b}$. Consequently, for

$$\widehat{\mathcal O}_k = \frac1N\operatorname{tr}M^k,$$

the planar connected $r$-point function scales as

$$\left\langle \widehat{\mathcal O}_{k_1}\cdots \widehat{\mathcal O}_{k_r} \right\rangle_{\mathrm c} = O\!\left(N^{2-2r}\right).$$

The frozen one-vertex checks are

$$\begin{array}{c|c|c|c|c} \text{pairing} & V & E & F & N^{V-E+F}\\ \hline \text{planar} & 1 & 2 & 3 & N^2\\ \text{crossed} & 1 & 2 & 1 & N^0 \end{array}$$

in the canonical matrix normalization. If the same written matrix field is instead assigned

$$\widetilde S_{\mathrm m} = \int\mathrm d^d x\, \operatorname{tr} \left[ \frac12(\partial M)^2+\frac{g}{4}M^4 \right]$$

with no overall $N$, propagators and vertices are both $O(1)$. The powers become $N^F$: $N^3$ for the planar pairing and $N$ for the crossed pairing. This is the required counterexample to normalization-blind topology counting.

The ribbon identity and its genus organization are the content of the double-line expansion ’t Hooft 1974, §§2–3, pp. 466–471. They organize powers of $N$; by themselves they do not construct a worldsheet measure, establish target-space locality, identify the complete spectrum, or prove a string dual.

Large-N scaling comparison

The table is a semantic counterpart to the figure and fixes the conventions reused throughout the chapter.

Large-N scaling, normalization, and validity comparison

Family or claim	Indices, action, and held data	Elementary count	Normalized observable and leading scaling	Required limit or failure test
O(N) vector	φᵢ; interaction λ(φᵢφᵢ)²/(4N); hold λ fixed	Propagator N⁰; vertex N⁻¹; closed index loop N; graph N^(L−V)	Q̂ = N⁻¹φᵢφᵢ = O(1); leading vacuum and cactus graphs are O(N)	Check L = V + 1 for the proposed leading family and verify that infrared or critical enhancements do not offset 1/N
Hermitian matrix	Mⁱⱼ; S = N tr V(M); hold all single-trace couplings gₖ fixed	Propagator N⁻¹; vertex N; face N; graph N^(V−E+F)	Ôₖ = N⁻¹ tr Mᵏ = O(1); a connected r-point function is O(N^(2−2r)) at genus zero	Verify W(z) ∼ 1/z, density normalization and positivity, support choice, and distance from a critical edge
Adjoint gauge	Aij; S = N/(2λt) ∫ tr F²; hold the renormalized λt = gYM2N fixed	Double-line propagator N⁻¹; interaction vertex N in the stated convention; adjoint face N	N⁻¹ tr F² and normalized Wilson loops are O(1); connected invariant correlators are suppressed	Fix gauge group, generator trace, matter representation, Nf scaling, state, regulator, and μ before counting
Fundamental-matter boundary	Fundamental color line with Nf fixed	Each quark loop removes an adjoint face and contributes Nf; relative weight Nf/N	A disk-like unnormalized boundary is N¹ rather than the adjoint sphere's N²	If Nf/N is fixed instead, use the Veneziano expansion; do not quote fixed-Nf suppression
Rank-three tetrahedral tensor	Tabc with O(N)3; vertex gN−3/2; hold g fixed	Propagator N⁰; vertex N⁻³ᐟ²; colored face N; graph N^(F−3V/2)	N−3TabcTabc = O(1); leading connected vacuum graphs scale as N3	Verify the invariant and covariance; melonic dominance is model-specific and is not a matrix-genus statement
Volume or orbifold equivalence	Compare only the common neutral sector at fixed ’t Hooft data	Factorization closes neutral loop equations only at leading N	Center-neutral, projection-invariant observables can agree at N = ∞	Center and projection symmetries must be unbroken; declare translation realization and the N, volume, lattice-spacing, and continuum order

For the vector row, a leading cactus graph has $L=V+1$ and therefore $N^{L-V}=N$. For the tensor row, inserting an elementary two-vertex melon adds three colored faces and two vertices:

$$\Delta\!\left(F-\frac32V\right) = 3-\frac32(2) =0.$$

The insertion preserves the leading power. This face count, rather than an Euler genus, is the elementary reason that the declared model selects melons Klebanov and Tarnopolsky 2017, §§2–3.

The vector, matrix, and gauge normalizations and their distinct leading free-energy powers are reviewed with explicit derivations in Mariño 2015, chs. 6–8, pp. 193–258.

Orders of limits are part of the result

A complete statement should display an order such as

$$\lim_{L\to\infty} \lim_{a\to0} \lim_{N\to\infty} \left\langle\widehat{\mathcal O}\right\rangle_{N,a,L},$$

or explain why another sequence is required. The notation does not assert that the limits commute.

Important obstructions include:

• Continuum and $N$. Bare couplings depend on the regulator and on $N$. Compare theories at fixed renormalized $\lambda(\mu)$ and a fixed physical scale, not at the same unrenormalized symbol.
• Volume and symmetry breaking. A finite-volume symmetric state can be a mixture of phases. Factorization and saddle selection can change if $N\to\infty$, $L\to\infty$, and a symmetry-breaking source is removed in another order.
• Infrared and $1/N$. A nominal $N^{-1}$ correction multiplied by an infrared-divergent integral need not remain small as $p\to0$ or a correlation length diverges.
• Matrix criticality. At fixed distance from a support transition, the genus expansion is ordered. Approaching the transition with $N$ can make all genera comparable and require a double-scaled variable.
• Volume reduction. Taking $N\to\infty$ at fixed reduced volume is useful only while the center and projection symmetries remain unbroken and only for neutral observables Kovtun, Ünsal, and Yaffe 2007, §§2–4.
• Exponentially small sectors. Effects of order $e^{-cN}$ vanish at every fixed order in $1/N$ but can govern level splitting, saddle competition, or large-order behavior.

A simple mathematical witness is

$$f_N(x)=\frac{1}{1+Nx}, \qquad x\ge0.$$

Then

$$\lim_{x\to0^+}\lim_{N\to\infty}f_N(x)=0, \qquad \lim_{N\to\infty}\lim_{x\to0^+}f_N(x)=1.$$

Pointwise large-$N$ control away from $x=0$ is not uniform near the boundary.

First application: vector versus ’t Hooft limits

The vector and gauge theories can both be called “large $N$,” but their surviving diagrams differ.

In the $O(N)$ vector model, keep $\lambda$ fixed in $\lambda(\phi_i\phi_i)^2/(4N)$. A connected leading vacuum graph with $V$ quartic vertices and $L=V+1$ index loops scales as $N$. The natural singlet is $N^{-1}\phi_i\phi_i$.

In $SU(N)$ gauge theory, keep $\lambda_{\mathrm t}=g_{\mathrm{YM}}^2N$ fixed. Adjoint propagators carry two oriented strands, and a connected planar vacuum graph scales as $N^2$ regardless of its number of vertices. The natural local invariant is $N^{-1}\operatorname{tr}F^2$. Holding $g_{\mathrm{YM}}$ rather than $\lambda_{\mathrm t}$ fixed makes $\lambda_{\mathrm t}$ grow with $N$ and does not define the same perturbative reorganization.

The different $N$ powers reflect different numbers of degrees of freedom—$O(N)$ versus $O(N^2)$—and different index combinatorics. Neither limit is more fundamental, and neither supplies the tensor scaling by analogy.

Common pitfalls

Writing only the coupling that is held fixed. The same symbol can accompany different field and action normalizations. State the quadratic term, propagator, vertex, index-loop factor, and normalized observable.

Calling every trace a boundary without tracking its prefactor. A topological boundary changes the Euler characteristic, while an explicit $1/N$ in a normalized trace changes the power again. Record both effects.

Reading a sharp large-$N$ transition at finite $N$. A finite stable matrix integral is analytic in its couplings. The sharp support transition belongs to the limiting density and is rounded in a critical finite-$N$ window.

Exercises

1. Count the planar and crossed one-vertex quartic matrix contractions in both matrix normalizations used above.

Solution

With $S=N\operatorname{tr}V(M)$, both graphs have $V=1$ and $E=2$. The planar pairing has $F=3$ and gives

$$N^{1-2+3}=N^2.$$

The crossed pairing has $F=1$ and gives $N^{1-2+1}=N^0$. Without the overall $N$, vertices and propagators are $N^0$, so only faces remain: the powers are $N^3$ and $N$.

2. A connected vector cactus graph has $V$ quartic vertices and $V+1$ closed index loops. Show that it contributes to the $O(N)$ free energy.

Solution

Every vertex supplies $N^{-1}$ and every loop supplies $N$. Hence

$$N^{L-V}=N^{(V+1)-V}=N.$$

Subleading vector graphs have fewer index loops at fixed $V$ and are suppressed.

3. An unnormalized planar one-boundary matrix amplitude scales as $N$. What is the scaling after inserting $\widehat{\mathcal O}=N^{-1}\operatorname{tr}M^k$?

Solution

The topological boundary gives $N^{2-b}=N$ for $b=1$. The explicit $N^{-1}$ in $\widehat{\mathcal O}$ changes this to $N^0$, consistent with an $O(1)$ one-point function.

4. Verify the two iterated limits of $f_N(x)=1/(1+Nx)$.

Solution

At any fixed $x>0$, $f_N(x)\to0$ as $N\to\infty$, and the subsequent $x\to0^+$ limit is zero. At fixed $N$, $f_N(x)\to1$ as $x\to0^+$, and the subsequent $N\to\infty$ limit remains one.

References

• Klebanov, I. R., and Tarnopolsky, G. (2017). “Uncolored Random Tensors, Melon Diagrams, and the Sachdev–Ye–Kitaev Models.” Physical Review D 95, 046004. doi:10.1103/PhysRevD.95.046004. Open PDF.
• Kovtun, P., Ünsal, M., and Yaffe, L. G. (2007). “Volume Independence in Large $N_c$ QCD-Like Gauge Theories.” Journal of High Energy Physics 2007(06), 019. doi:10.1088/1126-6708/2007/06/019. Open PDF.
• Mariño, M. (2015). Instantons and Large N: An Introduction to Non-Perturbative Methods in Quantum Field Theory. Cambridge University Press. doi:10.1017/CBO9781107705968.
• ’t Hooft, G. (1974). “A Planar Diagram Theory for Strong Interactions.” Nuclear Physics B 72, 461–473. doi:10.1016/0550-3213(74)90154-0.