  We study the large $N$ 't Hooft expansion of the chiral partition function of
2d $U(N)$ Yang-Mills theory on a torus. There is a long-standing puzzle that no
explicit holomorphic anomaly equation is known for the partition function,
although it admits a topological string interpretation. Based on the chiral
boson interpretation we clarify how holomorphic anomaly arises and propose a
natural anti-holomorphic deformation of the partition function. Our deformed
partition function obeys a fairly traditional holomorphic anomaly equation.
Moreover, we find a closed analytic expression for the deformed partition
function. We also study the behavior of the deformed partition function both in
the strong coupling/large area limit and in the weak coupling/small area limit.
In particular, we observe that drastic simplification occurs in the weak
coupling/small area limit, giving another nontrivial support for our
anti-holomorphic deformation.
