Large N Expansion in Matrix Models

1. The 't Hooft Limit: A New Expansion for Gauge Theory

As you know from your study of Yang-Mills theory, non-Abelian gauge theories are notoriously difficult to solve, especially at strong coupling where standard perturbation theory fails. The core problem is the lack of a small, dimensionless parameter to expand in.

In 1974, 't Hooft proposed a radical idea: consider an SU(N) gauge theory and take the number of colors, $N$, to be very large. This seems counter-intuitive, as one might expect a theory with $N^2-1$ gluons to become more complex as $N$ grows. However, 't Hooft showed that if you take the limit $N \to \infty$ while keeping a new parameter, the 't Hooft coupling $\lambda = g^2 N$, fixed, the theory simplifies. This is called the 't Hooft limit.

The underlying reason for this simplification is that the $N^2$ degrees of freedom are not independent; they are linked by the SU(N) symmetry. As $N$ increases, the collective dynamics of these fields become more "rigid" and, in a specific sense, classical.

To see how this works, let's look at the structure of Feynman diagrams in this limit.

The Double-Line Notation and 't Hooft Counting

As you've just read, the key to understanding the large $N$ limit is the double-line notation. Since the gluon field $A_\mu$ is an $N \times N$ matrix in the adjoint representation, we can think of it as carrying two indices, $(A_\mu)^i_j$, where $i$ transforms in the fundamental ($\mathbf{N}$) representation and $j$ transforms in the anti-fundamental ($\bar{\mathbf{N}}$).

This allows us to represent the gluon propagator not as a single curly line, but as two parallel lines with opposite-pointing arrows, representing the flow of the $i$ and $j$ indices.

• Propagator: A double line connects two vertices. From the action $S = -\frac{N}{2\lambda} \int \text{tr} F^2$ (with fields rescaled), the propagator scales as $\sim \frac{\lambda}{N}$.
• Vertex: A vertex joins propagators. The vertices scale as $\sim \frac{N}{\lambda}$.
• Index Loops: Every closed loop of a single color line (an "index loop") corresponds to a sum over the $N$ colors, $\sum_{i=1}^N \delta^i_i = N$.

The $N$-scaling of any vacuum Feynman diagram is therefore given by:

As shown in the notes, this counting rule reveals a remarkable topological organization. Diagrams that can be drawn on a 2D plane without lines crossing are called planar diagrams. Diagrams that require crossings (i.e., must be drawn on a surface with "handles" like a torus) are non-planar.

• Planar diagrams have a scaling that is independent of their complexity (in terms of loops), contributing at the leading order in $1/N$. For vacuum diagrams, this is $O(N^2)$.
• Non-planar diagrams are suppressed by factors of $1/N^2$ for each "handle" required to draw them. The simplest non-planar vacuum diagram scales as $O(N^0)$.

Therefore, in the strict $N \to \infty$ limit, we only need to sum the planar diagrams! While this is still an infinite sum, it's a monumental simplification.

2. Matrix Models: A Solvable Toy System

The 't Hooft counting is a general principle, but applying it to solve 4D Yang-Mills is still beyond our reach. To gain a more concrete understanding of the large N limit, we can turn to a much simpler "toy" quantum field theory: a zero-dimensional matrix model.

A matrix model is defined by a path integral over a single $N \times N$ Hermitian matrix $M$, rather than a field $A_\mu(x)$ over spacetime. The partition function is:

where $dM = \prod_{i,j} dM_{ij}$ and $W(M)$ is a polynomial potential, e.g., $W(M) = \frac{1}{2} M^2 + \frac{g_4}{4} M^4$.

This system has the same U(N) matrix structure as Yang-Mills, so the same topological expansion applies.

From Matrices to Eigenvalues

The two key insights from these notes are:

1. Topological Expansion: The free energy $\log Z$ has an expansion of the form:

   $$\log Z=\sum _{g=0}^{\infty }{N}^{2-2g}{F}_g(t)$$

   where $g$ is the genus (number of handles) of the Riemann surface on which the diagrams can be drawn. $g=0$ is the sphere (planar diagrams), $g=1$ is the torus, and so on. This confirms that the leading contribution at large $N$ comes from the planar $g=0$ sector.

2. Eigenvalue Reduction: Because the action and measure are invariant under U(N) conjugation ($M \to \Omega M \Omega^\dagger$), we can change variables from the $N^2$ matrix elements $M_{ij}$ to the $N$ eigenvalues $\lambda_i$ of $M$ and the $N^2-N$ angular variables of the U(N)\) group. The integral over the group factorizes, and the partition function becomes:

   $$Z\propto \int \prod _{i=1}^{N}d{\lambda }_i\exp \left(-\frac{N}{t}\sum _{i}W({\lambda }_i)\right)\prod _{j<k}({\lambda }_j-{\lambda }_k{)}^2$$

   This is an extraordinary simplification. We have reduced a problem with $N^2$ degrees of freedom to one with just $N$. The term $\prod (\lambda_j - \lambda_k)^2$ is the Vandermonde determinant, which arises as a Jacobian from the change of variables. It can be interpreted as a repulsive force between the eigenvalues.

The Classical Limit: Eigenvalue Distribution

The simplification continues. In the large $N$ limit, the $N$ discrete eigenvalues behave like a continuous fluid. We can rewrite the action in the exponent as:

Since $S_{\text{eff}}$ is large (proportional to $N^2$), the path integral is dominated by the configuration of eigenvalues that minimizes this effective action. This is the saddle-point approximation, and its validity at large N is why the theory is said to become classical.

We can define a continuous eigenvalue density $\rho(\lambda) = \frac{1}{N} \sum_i \delta(\lambda - \lambda_i)$. The problem then reduces to finding the function $\rho(\lambda)$ that minimizes a functional. For the simple Gaussian model ($W(M) = \frac{1}{2} M^2$), the solution is the famous Wigner semicircle law:

This is a beautiful and concrete demonstration of how the dynamics of a QFT simplify in the large N limit. An intractable sum over diagrams for $N^2$ variables becomes a solvable problem for a single density function.

3. Factorization and the String Theory Connection

This "classical" nature of the large $N$ limit also appears in correlation functions. Consider gauge-invariant, single-trace operators, like $\mathcal{O} = \frac{1}{N} \text{tr}(M^k)$ in matrix models or $\mathcal{O} = \frac{1}{N} \text{tr}(F_{\mu\nu} F^{\mu\nu})$ in gauge theory. A key result of 't Hooft counting is that connected correlation functions of such operators scale as:

For a two-point function, this means $\langle \mathcal{O}_1 \mathcal{O}_2 \rangle_c \sim N^0$. However, the disconnected part scales as $\langle \mathcal{O}_1 \rangle \langle \mathcal{O}_2 \rangle \sim N \cdot N = N^2$. Therefore, at large $N$:

This is called large N factorization. It means that quantum fluctuations of these operators are suppressed by $1/N^2$. This is another signature of a classical theory.

Finally, we arrive at 't Hooft's grandest vision.

The expansion of the gauge theory free energy, $\log Z = \sum_{g=0}^{\infty} N^{2-2g} F_g(\lambda)$, looks exactly like the genus expansion of the partition function in a string theory, where amplitudes are calculated by summing over all possible topologies of the string worldsheet.

This led 't Hooft to the stunning conjecture that SU(N) gauge theory in the large $N$ limit is secretly a string theory!

The mapping is:

• Gauge Theory $\leftrightarrow$ String Theory
• Expansion parameter $1/N$ $\leftrightarrow$ String coupling $g_s$
• 't Hooft coupling $\lambda = g^2 N$ $\leftrightarrow$ Related to string tension $1/\alpha'$

So, the N \to \infty limit of the gauge theory corresponds to the weakly coupled ($g_s \to 0$) limit of the dual string theory. This profound insight is the conceptual seed from which the AdS/CFT correspondence grew.

Conclusion

In this lesson, we have uncovered a hidden structure within gauge theories and matrix models that emerges in the large $N$ limit. This provides a new, non-perturbative way to analyze these strongly-coupled systems.

Key Takeaways:

• The 't Hooft limit ($N \to \infty$ with $\lambda = g^2 N$ fixed) provides a new expansion parameter, $1/N$, for SU(N) gauge theories.
• In this limit, Feynman diagrams are organized by topology. The leading contributions come from planar diagrams, which can be drawn on a sphere, while non-planar diagrams are suppressed by powers of $1/N^2$.
• The large $N$ limit is a classical limit. Dynamics simplify dramatically: correlation functions of single-trace operators factorize, and in solvable matrix models, the $N^2$ degrees of freedom collapse into a single eigenvalue density function.
• The topological expansion of large $N$ theories is mathematically equivalent to the perturbative expansion of a dual string theory, with the string coupling $g_s \sim 1/N$.

Next Lesson Preview:

We have ended on a tantalizing cliffhanger: large $N$ gauge theories are string theories. But a string theory must live in some spacetime. What is the nature of this spacetime, and how does it relate to the gauge theory we started with? In our next lesson, we will explore the holographic principle, which states that the information in a volume of space can be encoded on its boundary. This will lead us to the geometry of Anti-de Sitter space and the precise formulation of the AdS/CFT correspondence.