Instanton partition functions in N=2 SU(N) gauge theories with a general surface operator, and their W-algebra duals

We write down an explicit conjecture for the instanton partition functions in 4d N = 2 SU(N) gauge theories in the presence of a certain type of surface operator. These surface operators are classified by partitions of N, and for each partition there is an associated partition function. For the partition N = N we recover the Nekrasov formalism, and when N = 1+...+1 we reproduce the result of Feigin et. al. For the case N = 1 + (N-1) our expression is consistent with an alternative formulation in terms of a restricted SU(N) x SU(N) instanton partition function. When N - 1+...+1+2 the partition functions can also be obtained perturbatively from certain W-algebras known as quasi-superconformal algebras, in agreement with a recent general proposal.