Exact 1/N expansion of Wilson loop correlators in $\mathcal{N}=4$ Super-Yang-Mills theory

Supersymmetric circular Wilson loops in $\mathcal{N}=4$ Super-Yang-Mills theory are discussed starting from their Gaussian matrix model representations. Previous results on the generating functions of Wilson loops are reviewed and extended to the more general case of two different loop contours, which is necessary to discuss coincident loops with opposite orientations. A combinatorial formula representing the connected correlators of multiply wound Wilson loops in terms of the matrix model solution is derived. Two new results are obtained on the expectation value of the circular Wilson loop, the expansion of which into a series in $1/N$ and to all orders in the 't~Hooft coupling $\lambda$ was derived by Drukker and Gross about twenty years ago. The connected correlators of two multiply wound Wilson loops with arbitrary winding numbers are calculated as a series in $1/N$. The coefficient functions are derived not only as power series in $\lambda$, but also to all orders in $\lambda$ by expressing them in terms of the coefficients of the Drukker and Gross series. This provides an efficient way to calculate the $1/N$ series, which can probably be generalized to higher-point correlators.