Ivancevic_Applied-Diff-Geom

Applied Bundle Geometry 787Let us also note that, although the WDVV equations can be fulfilledonly for some specific choices of the coordinates a i on the moduli space,they still admit any linear transformation. This defines the flat structureson the moduli space, and we often call a i flat coordinates.4.14.12.2 Perturbative SW PrepotentialsBefore going into the discussion of the WDVV equations for the completeSW prepotentials, let us note that the leading perturbative part of themshould satisfy the equations (4.290) by itself (since the classical quadraticpiece does not contribute into the third derivatives). On the other hand,if the WDVV equations are fulfilled for perturbative prepotential, it is anecessary condition for them to hold for complete prepotential.To determine the one–loop perturbative prepotential from the field theorycalculation, let us note that there are two origins of masses in N = 2SUSY YM models: first, they can be generated by vacuum values of thescalar φ from the gauge super–multiplet. For a super–multiplet in representationR of the gauge group G this contribution to the prepotential isgiven by the analog of the Coleman-Weinberg formula (from now on, weomit the classical part of the prepotential from all expressions):F R = ± 1 4 Tr R φ 2 log φ, (4.294)and the sign is ‘+’ for vector super–multiplets (normally they are in theadjoint representation) and ‘−’ for matter hypermultiplets. Second, thereare bare masses m R which should be added to φ in (4.294). As a result,the general expression for the perturbative prepotential isF = 1 4− 1 4∑vectormplets∑hypermpletsTr A (φ + M n I A ) 2 log(φ + M n I A ) −Tr R (φ + m R I R ) 2 log(φ + m R I R ) + f(m), (4.295)where the term f(m), depending only on masses, is not fixed by the (perturbative)field theory, but can be read off from the non-perturbative description,and I R denotes the unit matrix in the representation R.As an example, consider the SU(n) gauge group. Then, e.g., perturba-