Ivancevic_Applied-Diff-Geom

Applied Bundle Geometry 789of the type (a i ± b j ), where moduli a i and b j are either periods ormasses 38 . This is the case for the models that contain either massivematter hypermultiplets in the first fundamental representation (or itsdual), or massless matter in the square product of those. Troubles arisein all other situations because of the terms with a i ± b j ± c k ± . . ..• At value r = 2, like a i ’s lying in irrep of G, masses m α ’s can be regardedas lying in irrep of some ˜G so that if G = A n , C n , D n , ˜G = An , D n ,C n accordingly. This correspondence ‘explains’ the form of the massterm in the prepotential f(m).4.14.12.3 Associativity ConditionsIn the context of the 2D LG topological theories, the WDVV equationsarose as associativity condition of some polynomial algebra. Mironov hasproved in [Mironov (1998)] that the equations in the SW theories have thesame origin.In this case, one deals with the chiral ring formed by a set of polynomials{Φ i (λ)} and two co-prime (i.e., without common zeroes) fixed polynomialsQ(λ) and P (λ). The polynomials Φ form the associative algebra with thestructure constants Cij k given with respect to the product defined by moduloP ′ :the associativity condition beingΦ i Φ j = C k ijΦ k Q ′ + (∗)P ′ −→ C k ijΦ k Q ′ , (4.296)(Φ i Φ j ) Φ k = Φ i (Φ j Φ k ) , (4.297)i.e., C i C j = C j C i , (C i ) j k = Cj ik . (4.298)Now, in order to get from these conditions the WDVV equations, one needsto choose properly the flat moduli:a i = −n ( )i(n − i) Res P i/n dQ , n = ord(P ).Then, there exists the prepotential whose third derivatives are given by theresidue formulaF ijk = 12πi Res Φ i Φ j Φ kP ′ =0 P ′ . (4.299)38 This general rule can be easily interpreted in D–brane terms, since the interactionof branes is caused by strings between them. The pairwise structure (a i ± b j ) exactlyreflects this fact, a i and b j should be identified with the ends of string.