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We write down an explicit conjecture for the instanton partition functions in4d N=2 SU(N) gauge theories in the presence of a certain type of surfaceoperator. These surface operators are classified by partitions of N, and foreach partition there is an associated partition function. For the partition N=Nwe recover the Nekrasov formalism, and when N=1+...+1 we reproduce the resultof Feigin et. al. For the case N=1+(N-1) our expression is consistent with analternative formulation in terms of a restricted SU(N)xSU(N) instantonpartition function. When N=1+...+1+2 the partition functions can also beobtained perturbatively from certain W-algebras known as quasi-superconformalalgebras, in agreement with a recent general proposal.

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