Wilson loops in large symmetric representations through a double-scaling limit

We derive exact formulas for circular Wilson loops in the 𝒩 = 4 and 𝒩 = 2 * theories with gauge groups U( N ) and SU( N ) in the k -fold symmetrized product representation. The formulas apply in the limit of large k and small Yang-Mills coupling g , with fixed effective coupling κ ≡ g 2 k , and for any finite N . In the SU(2) and U(2) cases, closed analytic formulas are obtained for any k , while the 1 /k series expansions are asymptotic. In the N ≫ 1 limit, with N ≪ k , there is an overlapping regime where the formulas can be confronted with results from holography. Simple formulas for correlation functions between the k -symmetric Wilson loops and chiral primary operators are also given.