large N expansion

From Physics wiki

A large N expansion is effectively a genus expansion in Feynman diagrams such that as the rank $N\,$ of the gauge group is increased, the contribution from higher genus diagrams are suppressed. At $N \to \infty\,$ , only planar diagrams contribute, while at finite $N\,$ there are $\tfrac{1}{N}\,$ corrections.

1 Yang-Mills theory
2 't Hooft's double-line notation
3 Eguchi-Kawai reduction
4 Connection with string theory
5 References

Yang-Mills theory

The 't Hooft coupling for Yang-Mills theory is defined as

$\lambda = g_{YM}^2 N\,$ .

't Hooft's double-line notation

Observe that

$\frac{g^2}{2} \operatorname{tr}\left( [A_\mu A_\nu] [A^\mu A^\nu] \right)\,$	$= \frac{g^2}{2} \operatorname{tr}\left(A_\alpha A_\beta A_\gamma A_\delta \right)\left(2 \eta^{\alpha\gamma}\eta^{\beta\delta} - 2\eta^{\alpha\delta} \eta^{\beta\gamma}\right)\,$ ,
	$= \frac{g^2}{2} \operatorname{tr}\left(A_\alpha A_\beta A_\gamma A_\delta \right)\left( 2\eta^{\alpha\gamma}\eta^{\beta\delta} - \eta^{\alpha\beta} \eta^{\gamma\delta} - \eta^{\beta\gamma} \eta^{\alpha\delta}\right)\,$ (after symmetrizing over cyclic permutations).