We look into the formula, due to Douglas and Dijkgraaf,
on the generating function, F_g(q), of the number of
simply ramified covers of genus g>=1 over a fixed
elliptic curve with marked points. Their result is that
F_g(q) is a quasimodular form of weight 6g6 on the full modular
group PSL_2(Z).
There are two ways of computing F_g(q): the fermionic
count and the bosonic count. The fermionic counting
is a mathematical treatment, and we will give a mathematical
proof to the formula. On the other hand, the bosonic counting
rests on physical arguments, which involves path integrals
on trivalent Feynman diagrams. We will compute F_g(q) for small genera
with bosonic count.
