  Recently, Okuyama and Sakai proposed a novel holomorphic anomaly equation for
the partition function of 2d Yang-Mills theory on a torus, based on an
anholomorphic deformation of the propagator in the bosonic formulation. Using
the boson-fermion correspondence, we derive the formula for the deformed
partition function in fermionic description and give a proof of the holomorphic
anomaly equation.
