We prove the Bershadsky-Cecotti-Ooguri-Vafa's conjecture for all genus Gromov-Witten potentials of the quintic threefolds, by identifying the Feynman graph sum with the $\nmsp$ stable graph sum via an R-matrix action. The Yamaguchi-Yau functional equations and the formulas of $F_1,F_2$, are direct consequences of the BCOV Feynman sum rule.

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We show that the angles between Teichmüller geodesic rays issuing from a common point, defined by using the law of cosines, do not always exist. The proof uses an estimation for the Teichmüller distance on finite dimensional Teichmüller spaces. As a consequence, the Teichmüller space equipped with the Teichmüller metric is not a CAT(k) space for any k∈R . We also discuss some necessary conditions for the existence of angle between the Teichmüller geodesics.