Ivancevic_Applied-Diff-Geom

784 Applied Differential Geometry: A Modern IntroductionNote that the left hand side can be interpreted in the gauge theory as arenormalization group equation.For simple laced g, the curves R(k, z) = 0 are modular invariant. Physically,the gauge theories for these Lie algebras are self–dual. For nonsimply–laced g, the modular group is broken to the congruence subgroupΓ 0 (2) for g = B n , C n , F 4 , and to Γ 0 (3) for G 2 . The Hamiltonians of thetwisted CM–systems for non–simply laced g are also transformed underLanden transformations into the Hamiltonians of the twisted CM–systemfor the dual algebra g ∨ . It would be interesting to determine whether suchtransformations exist for the spectral curves or the corresponding gaugetheories themselves.4.14.12 SW Theory and WDVV EquationsAs presented above, N. Seiberg and E. Witten proposed in [Seiberg andWitten (1994a); Seiberg and Witten (1994b)] a new way to deal with thelow–energy effective actions of N = 2 4D supersymmetric gauge theories,both pure gauge theories (i.e., containing only vector super–multiplet) andthose with matter hypermultiplets. Among other things, they have shownthat the low–energy effective actions (the end–points of the renormalizationgroup flows) fit into universality classes depending on the vacuum of thetheory. If the moduli space of these vacua is a finite–dimensional variety,the effective actions can be essentially described in terms of system withfinite–dimensional phase space (number of degrees of freedom is equal tothe rank of the gauge group), although the original theory lives in a many–dimensional space–time.4.14.12.1 WDVV EquationsNow, it turns out that the prepotential of SW effective theory satisfies thefollowing set of Witten–Dijkgraaf–Verlinde–Verlinde (WDVV) equations:F i F −1kF j = F j F −1k F i (i, j, k = 1, . . . , n), (4.290)where F i are the matrices on a moduli space M of third order derivatives∂ 3 F(F i ) jk =∂a i ∂a j ∂a kof a function F(a 1 , . . . , a n ), with the prepotential variables a i .