  It has recently been argued by Alday et al that the inclusion of surface
operators in 4d N=2 SU(2) quiver gauge theories should correspond to insertions
of certain degenerate operators in the dual Liouville theory. So far only the
insertion of a single surface operator has been treated (in a semi-classical
limit). In this paper we study and generalise this proposal. Our approach
relies on the use of topological string theory techniques. On the B-model side
we show that the effects of multiple surface operator insertions in 4d N=2
gauge theories can be calculated using the B-model topological recursion
method, valid beyond the semi-classical limit. On the mirror A-model side we
find by explicit computations that the 5d lift of the SU(N) gauge theory
partition function in the presence of (one or many) surface operators is equal
to an A-model topological string partition function with the insertion of (one
or many) toric branes. This is in agreement with an earlier proposal by Gukov.
Our A-model results were motivated by and agree with what one obtains by
combining the AGT conjecture with the dual interpretation in terms of
degenerate operators. The topological string theory approach also opens up new
possibilities in the study of 2d Toda field theories.
