Questions tagged [large-n]

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Isn't AdS/CFT an end to String theory as a fundamental theory?

I start with the Large $N$ QCD paper by 't Hooft. When 't Hooft published his paper on Large $N$ QCD it was clear why the string theory of hadrons due to Gabriele Veneziano could make sense. But at ...
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2 votes
1 answer
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Relationship between multiplicity in the $k$-fold product of fundamentals and irrep dimension at large $N$

Equation 3.5 of this paper by Gross and Klebanov makes the following interesting claim. Take a group $U(N)$, with $N$ large, and consider the reducible representation $\mathcal{H}_{fund}^{\otimes k}$ ...
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3 votes
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Does planar (or non-planar) ${\cal N}=4$ SYM contain bound states?

Does planar/large number of colors ${\cal N}=4$ SYM contain bound states at strong or weak coupling? Are there bound states in the non-planar limit at strong or weak coupling?
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Loop integral in $d$ dimensions

I am studying large $N$ Quantum Field Theory and I am having a hard time calculating the expansion of the simple loop integral of eq.(13.123) of Peskin and Schroeder. $$ \int\frac{d^dk}{(2\pi)^d}\frac{...
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1 vote
1 answer
144 views

What is the purpose and meaning of taking the 't Hooft parameter to infinity?

I am following Hong Liu's MIT 8.821 String Theory and Holographic Duality lectures. He starts discussing the large-$N$ expansion in the context of a hermitian matrix model described by the Lagrangian $...
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Resources on calculation of beta function for $\mathrm{O}(N)$ model

Is there any useful introductory material about the calculation of the beta function in $\phi^4$ scalar theory and $O(N)$ model? I would also like to generalize this to the large-$N$ limit.
3 votes
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Beta function in the large $N$ limit

I am currently studying Quantum Field Theory in the large $N$ limit (https://arxiv.org/abs/hep-th/9601080) and I am trying to understand how to calculate RG $β$-function $N$ β-function" /> How can one ...
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0 votes
2 answers
112 views

$O(N)$ symmetry in three dimensions

Recently, In a research article on magnetism, I came across the term "$O(N)$ symmetry for three dimensions with the limit $N->infinity$". What does it mean? When I tried to search about ...
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How large is large $N$?

I once heard Lenny Susskind relate the question: "how many particles do you need in a box for the ideal gas law to 'pretty much' hold?" Obviously this question requires a notion of 'pretty ...
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Quark propagator

I was reading about Large N QCD, and specifically, T Hooft Double line notation, when I stumbled across the quark propagator- $$\langle \psi^a (x) \overline{\psi^b}(y) \rangle = \delta^{ab} S(x-y) $$ ...
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Graviton fluctuation suppressed in large $N$ matters

I have a question about semiclassical gravity approximation. For probing Hawking radiation, we usually treat gravitational theory as semiclassical assuming large $N$ matters. However, I do not know ...
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How can I show that $1/N$ expansion for large $N$ matrix models have a string theoretical perturbation expansion?

While surfing through some further reading suggestions on string theory, I stumbled upon this slide from a talk by Nathan Seiberg. I wanted to derive the main argument by applying a perturbation ...
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5 votes
1 answer
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What's the difference and relations between $SU(N)$ Schwinger boson and $CP(N\!-\!1)$ non-linear sigma model?

There are two ways when dealing with spin system(Heisenberg model): non-linear $\sigma$ model and Schwinger boson. Non-linear $\sigma$ model When taking large $S$ limit, the quantum fluctuation of ...
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4 votes
0 answers
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How do I show that a given Wilson loop satisfies the loop equation?

In the book Methods of Contemporary Gauge Theory by Yuri Makeenko, the loop equation in the large-$N$ limit is given by $$\partial^x_\mu \frac{\delta}{\delta \sigma_{\mu \nu}} W(C) = \lambda \oint_C ...
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4 votes
2 answers
456 views

How to know if a Feynman diagram is planar?

A planar diagram is defined as being one of the leading diagrams for $N \to \infty$ (large $N$ expansion), and, as I understand it, it should have the lowest genus when compared to a non-planar ...
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1 vote
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Why is the self-energy for quarks in $d=2$ Large $N$ QCD only order $g^2$?

In an interesting article by 't Hooft , he is able to find the exact quark propagator, in the large $N$ limit of QCD. He finds that the full 1PI self-energy is given by: $$\Gamma(p)=-\frac{g^2}{2\pi} \...
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2 answers
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Besides instantons and large-$N$ what are some other general non-perturbative methods for quantum field theory?

Besides large-$N$, instantons, lattice QFT, what are some other non-perturbative methods that help us better understand QFTs like the large distance dynamics of Yang Mills and QCD?
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1 vote
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Question about the Dyson-Schwinger equation for 4-point function of the SYK model

In studying the SYK model, I have no idea how to get the kernel $K(t_a,t_b,t_3,t_4)$ and $\Gamma_0(t_1,t_2,t_3,t_4)$ and also ladder diagrams in the Dyson-Schwinger equation for 4 point function of ...
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1 answer
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Large-N of non-linear sigma model

Consider the partition function of non-linear sigma model is: $$Z=\int\mathcal{D}[\textbf n]\mathcal{D}[\lambda]\exp(-S)\\S=\frac{1}{f} \int d^dx \int d \tau [(\partial_\tau \textbf{n})^2+c^2(\nabla_x ...
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6 votes
1 answer
366 views

Exact solution of SU($N$) model in large $N$ limit

There is the statement about $\text{U}(N)$ model: It is not possible to solve $\text{U}(N)$ model even in large $N$ limit I do not understand this statement. If I go to large $N$ limit I know that ...
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3 votes
1 answer
147 views

Electrons with disorder & something like AdS/CFT duality

I know that consideration of electrons with disorder can be based on Feynman diagrams with disorder lines. In this approach, only non-crossing diagrams are important and give contribution to self-...
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2 votes
0 answers
67 views

Do large $N$ free fermion or WZW theories have a holographic dual in $AdS_3/CFT_2$?

I was wondering if for $N$ free Dirac fermions (or equivalently by bosonization, $N$ free bosons or an $SU(N)_1$ WZW theory plus an extra boson) have a holographic dual description via $AdS_3/CFT_2$? ...
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Is the leading order contribution to the double-trace operator anomalous dimension always $O(1/N^2)$?

Is the leading order contribution to the double-trace operator anomalous dimension always $O(1/N^2)$ ? I noticed that the double-trace contribution in Polchinski's paper hep-th/0907.0151 gets an ...
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1 vote
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Understanding some lines from 't Hooft's paper on large-N QCD in 1+1d

In 't Hooft's paper "A two-dimensional model for mesons", the author shows that two-dimensional (1+1) QCD in the large-N limit interestingly gives a theory of mesons. 't Hooft calculates the "mesonic ...
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0 votes
2 answers
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Question on the $1/N$ expansion

My question is from Coleman's Aspect of Symmetry, on the large $N$ approximation. We will consider the $O(N)$ version of the $\phi^4$ theory. Its Lagrangian density is given by: $$ \mathcal{L}=\...
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5 votes
1 answer
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Different purposes for using the Large-$N$ Expansion

I've started studying the Large-$N$ expansion and there seems to be several different reasons for using it. In the context of the SYK model, the limit is useful because it reorganizes the Feynman ...
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1 vote
1 answer
125 views

Difference between mean-field theory and large $N$ of $CP^{N-1}$

I am reading the Ch.14 of Auerbach, Interaction electrons and quantum magnetism about $CP^{N-1}$ which describes non-linear $\sigma$ model. The complex field is $\mathbf{z}=\left(z_{1}, z_{2}, \ldots,...
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  • 1,472
1 vote
0 answers
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How do we understand the results of $1/N$ or $\epsilon$ expansion beyond leading orders?

When we do $1/N$ expansions in, say, 2+1$D$ $O(N)$ models and try to extract all kinds of critical exponents from it, we get the following results for the scaling dimensions of various operators up to ...
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4 votes
1 answer
421 views

Is there confinement in the Gross-Neveu model?

The Gross-Neveu model is a simple quantum field theory in 2 space-time dimensions that is considered a toy model for QCD, in the sense that it realizes asymptotic freedom, chiral symmetry breaking ...
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2 votes
1 answer
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How does gauge theory become strongly coupled at large $N$ whether it's coupling is proportional to 1/$\sqrt N$?

At large $N$, gravity theory becomes weakly coupled is correct as we see from its formula that string's coupling is proportional to $1/N^2$. then how could gauge theory become strongly coupled?
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3 votes
1 answer
310 views

Manipulations with Traces: Saddle point integration in Large-$N$ model

For reference I am trying to work out the derivation in this paper, in which the partition function for an Ising model is approximated by replacing the Ising variables $\sigma_i$ with $N$ component ...
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3 votes
0 answers
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How to get the eigenvalue density contribution $\rho_1(x)$?

I'm studying the $1/N$ expansion beyond the planar limit in matrix models. Currently I'm trying to understand and reproduce the results of: Antisymmetric Wilson loops in $\mathcal N \geq 4$ SYM ...
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1 vote
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What physics describes a $SU(N)$ theory with large $N$?

Suppose that we have a system that can be described by $N$ (is a large number!) degree of freedom besides well-known degree of freedom like energy, momentum, spin, etc.. Now scattering processes are ...
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1 vote
1 answer
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Solving the $CP^N$ model in large $N$ limit

I have trouble filling in the essential step of solving the $CP^N$ model in large $N$ limit, described on Page 84 to 86 of Michael Dine's Supersymmetry and String Theory. The Lagrangian is given by $$...
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0 votes
1 answer
244 views

$1/N$ expansion in $SU(N)$ Yang-Mills theories at large $N$

It is a well-known fact that correlators of $SU(N)$ Yang-Mills theories at large $N$ are expanded in powers of the 't Hooft coupling $g_{YM}^{2}N$ and $1/N$. Is there a reason why an expansion in ...
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6 votes
1 answer
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A question on large-N limit?

Let's take $SU(N)$ for an example. The Lagrangian is $$\mathcal{L}=-\frac{1}{4g_{YM}^2}F_{\mu\nu}F^{\mu\nu}.$$ We can define the t'Hooft coupling as $$\lambda=g_{YM}^2N.$$ Then the large-$N$ limit ...
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3 votes
0 answers
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What is the meaning of the existence of a large $N$ limit in QFT?

Large $N$ limits are present in many different contexts: matrix models, gauge theories in various dimensions, conformal field theories (where $N$ is essentially the central charge). We often hear ...
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1 vote
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Non-renormalization theorems at large N

I know of numerous examples of non-renormaliation theorems in theories with SUSY - e.g. the non-renormaliation of the superpotential in 4d theories. However, I've never seen a non-renormaliation ...
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2 votes
1 answer
902 views

The logic of large $N$ expansion

I have some understanding of how the large-$N$ expansion works but feel like I'm missing the most important concepts. For example, I understand that in QCD the order of the diagram in $N$ depends ...
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1 vote
0 answers
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Central Charge of large $N$ Gauge theory in 't Hooft limit

It is well known that large N gauge theory in t'Hooft limit has central charge ~ $N^2$ I want to convince myself in this by considering simple example of: 1 flavor(meaning that we have only one ...
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2 votes
1 answer
374 views

Relation between $1/N$ and perturbative expansions in QFT

I heard many times phrases like "large $N$ is a clever way to organize the diagrammatic expansion" or "each diagram in the large $N$ expansion contains an infinite number of usual perturbative ...
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5 votes
2 answers
541 views

Large $N$ expansion vs AdS/CFT

I'm beginning to learn AdS/CFT and I have an elementary question. It is said that since it is very hard to calculate 4 point and above correlators in a strongly coupled CFT, we can use instead the ...
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4 votes
0 answers
273 views

Some questions about QCD [closed]

About QCD, I have two questions. I know I should propose one question one time, but they are actually two steps of the same question: Non-perturbative aspects of QCD. 1, Why do we need to solve QCD ...
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45 votes
0 answers
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$\operatorname{O}(N)$ sigma model at large $N$

I would like to better understand the main principles of large-$N$ expansion in quantum field theory. To this end I decided to consider simple toy-model with lagrangian (from Wikipedia) $ \mathcal{L} ...
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5 votes
0 answers
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Relation between holography and matrix models

Let's consider a 0-dimensional $N \times N$ Hermitean one matrix model. It is defined by a potential V(M). Its partition function is $$Z = \int_{H_{N}} dM e^{-\frac{1}{g}V(M)}$$ where $H_{N}$ is the ...
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  • 184
6 votes
1 answer
126 views

Normalization of Source Terms in Large-N Gauge Theory

Typically when you do the counting for large N gauge theory, you rescale fields so that the Lagrangian takes the form \begin{equation} \mathcal{L}=N[-\frac{1}{2g^2}TrF^2+\bar{\psi}_i\gamma^\mu D_\mu \...
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9 votes
2 answers
430 views

Large-$N$ Yang Mills

I've bumped into the study of the $SU(N)$ theory in the large-$N$ limit. I'm wondering in which way the study of this Yang-Mills theory, can give contribution to QCD with gauge group $SU(3)$, i.e. $N=...
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1 vote
0 answers
110 views

Large-N critical NLSM (equation 13.115 of Peskin and Schroeder)

Any opinions if the equation 13.115 of Peskin and Schroeder is true on arbitrary manifolds in arbitrary dimensions for the same Lagrangian? I a priori see no problem. The point I also want to ask is -...
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  • 4,439
5 votes
0 answers
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Some questions about the large-N Gross-Neveu-Yukawa model

Consider the following action with a fermionic field $\psi$ and a scalar field $\sigma$, $S = \int d^dx \{ -\bar{\psi}(\gamma^\mu \partial_\mu +\sigma )\psi + \Lambda^{d-4}[ \frac{(\partial_\mu \...
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  • 4,439
2 votes
1 answer
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Large-$N$ factorization of single-trace operators

Does anyone know where I can find a pedagogical explanation of large-$N$ factorization in $SU(N)$ gauge theories or nonlinear $O(N)$ sigma models (in the latter case the trace corresponds to a dot ...