# BCOV's holomorphic anomaly equation at genus one

+ 3 like - 0 dislike
87 views

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?

This post imported from StackExchange MathOverflow at 2015-10-23 17:29 (UTC), posted by SE-user Ahsan

 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\hbar$ysicsOverflo$\varnothing$Then drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.