# BCOV's holomorphic anomaly equation at genus one

BCOV in their famous paper (http://arxiv.org/abs/hep-th/9309140) state the genus one holomorphic anomaly equation (on page 53) to be $$\partial_i \partial_{\bar{j}} F_{1} = \frac{1}{2}C_{ikl}\bar{C}^{kl}_{\bar{j}} + (1-\frac{\chi}{24})G_{i \bar{j}}$$ where $C_{ijk}$ is the 3 point function or Yukawa coupling on Calabi-Yau moduli space, $$\bar{C}^{kl}_{\bar{j}} = e^{2K}\bar{C}_{\bar{a}\bar{b}\bar{j}}G^{k \bar{a}} G^{l \bar{c}},$$ $K$ being the Kahler potential for the Weil-Pietersson metric on CY moduli space.

In an earlier paper (http://arxiv.org/pdf/hep-th/9302103v1.pdf) on page 14 they claim that $$F_1 = \text{log}\big[\text{exp}(3+h^{1,1} -\frac{\chi}{12})K \,\, \text{det}[G^{-1}] |f(z)|^2\big]$$ with $f(z)$ a holomorphic function.

However this does not seem to check out. Taking a holomorhic and anti-holomorphic derivative on the above solution and using the "special geometry relation" (a relation expressing the curvature in terms of the metric and 3-point function) I get a similar equation, but not exactly the same as the BCOV equation above. In particular, there seems to be no way to get rid of the appearance of the Hodge number $h^{1,1}$. What seems to be the problem?

There is an introduction to the BCOV theory from math perspective in this artcle by Kanazawa and Zhou. In the case $h^{1,1}=1$, it is checked in page 16 that the $\mathcal{F}_1$ (non-holomorphic object) satisfies the anomaly equation. The general case should follow in a similar manner.