Ivancevic_Applied-Diff-Geom

786 Applied Differential Geometry: A Modern Introductionand the matrices F i (the third derivatives of the prepotential) are⎛ ⎞0 0 1F 1 = ⎝ 0 1 0 ⎠ ,⎛ ⎞0 1 0F 2 = ⎝ 1 0 0 ⎠ ,⎛ ⎞1 0 0F 3 = ⎝ 0 0 q ⎠ .1 0 00 0 q0 q 0The second example is the quantum cohomologies of CP 2 . In this case,the prepotential is given by the formulaF = 1 2 a 1a 2 2 + 1 2 a2 1a 3 +∞∑k=1N k a 3k−13(3k − 1)! eka2 , (4.292)where the coefficients N k (describing the rational Gromov–Witten classes)count the number of the rational curves in CP 2 and are to be calculated.Since the matrices F i have the form⎛ ⎞ ⎛⎞ ⎛⎞0 0 10 1 01 0 0F 1 = ⎝ 0 1 0 ⎠ , F 2 = ⎝ 1 F 222 F 223⎠ , F 3 = ⎝ 0 F 223 F 233⎠ ,1 0 00 F 223 F 233 0 F 233 F 333the WDVV equations are equivalent to the identity [Mironov (1998)]F 333 = F 2 223 − F 222 F 233 ,which, in turn, results into the recurrent relation defining the coefficientsN k :N k(3k − 4)! = ∑a+b=ka 2 b(3b − 1)b(2a − b)N a N b .(3a − 1)!(3b − 1)!The crucial feature of the presented examples is that, in both cases, thereexists a constant matrix F 1 . One can consider it as a flat metric on themoduli space. In fact, in its original version, the WDVV equations havebeen written in a slightly different form, that is, as the associativity conditionof some algebra. Having distinguished the (constant) metric η ≡ F 1 ,one can naturally rewrite (4.290) as the equationsC i C j = C j C i (4.293)for the matrices (C i ) jk≡ η −1 F i , i.e., C j ik = ηjl F ilk . Formula (4.293)is equivalent to (4.290) with j = 1. Moreover, this particular relationis already sufficient to reproduce the whole set of the WDVV equations(4.290).