# 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

$$\partial^x_\mu \frac{\delta}{\delta \sigma_{\mu \nu}} W(C) = \lambda \oint_C dy_\nu \; \delta^{(d)}(x-y)W(C_{yx})W(C_{xy}). \tag{12.59}$$

Equation (12.59) in the book, page 264. Where $$\frac{\delta}{\delta \sigma_{\mu \nu}}$$ is the area derivative and $$W(C)$$ is the expectation value of a closed wilson loop.

I am trying to understand this equation by evaluating an explicit example, but I couldn't find any case that I was able to evaluate (mostly because I don't understand the equation and how to apply the definitions, like the area derivative to the Wilson loop, the book doesn't have a particular case explaining these things).

For example, if we consider a circular Wilson loop with radius $$R$$ and expectation value given by $$e^{R^2}$$. I would like to know how do I take the area derivative of this and what $$W(C_{xy})W(C_{yx})$$ would correspond to in this case. Or maybe other example that could help me understand those abstract concepts.

• The first thing to do with such a formula is always to unpack the definitions. What definition does the book give for the "area derivative"? What difficulty do you have in applying it? – ACuriousMind May 23 at 10:54
• The definition of the area derivative is in equation 12.40, for example. – Slayer147 May 23 at 14:56
• The natural setting to study this is the strong coupling limit of lattice gauge theory, see for example, D. Friedan, “Some Nonabelian Toy Models in the Large N Limit,” Commun. Math. Phys. 78, 353 (1981). – Thomas Jun 11 at 15:27