Ivancevic_Applied-Diff-Geom

788 Applied Differential Geometry: A Modern Introductiontive prepotential for the pure gauge theory acquires the formF pertV= 1 4∑(a i − a j ) 2 log (a i − a j ) .ijThis formula states that when eigenvalues of the scalar fields in the gaugesuper–multiplet are non-vanishing (perturbatively a r are eigenvalues of thevacuum expectation matrix 〈φ〉), the fields in the gauge multiplet acquiremasses m rr ′ = a r − a r ′ (the pair of indices (r, r ′ ) label a field in the adjointrepresentation of G). In the SU(n) case, the eigenvalues are subjectto the condition ∑ i a i = 0. Analogous formulas for the adjoint mattercontribution to the prepotential isF pertA= − 1 4∑(a i − a j + M) 2 log (a i − a j + M) ,ijwhile the contribution of the fundamental matter readsF pertF= − 1 ∑(a i + m) 2 log (a i + m) .4iThe perturbative prepotentials have the following characteristics[Mironov (1998)]:(i) The WDVV equations always hold for the pure gauge theories:F pert = F pertV.(ii) If one considers the gauge super–multiplets interacting with the matterhypermultiplets in the first fundamental representation with masses m αF pert = F pertV+ rF pertF+ Kf F (m) (where r and K are some undeterminedcoefficients), the WDVV equations do not hold unlessK = r 2 /4, f F (m) = 1 ∑(m α − m β ) 2 log (m α − m β ) ,4α,βthe masses being regarded as moduli (i.e., the equations (4.290) contain thederivatives with respect to masses).(iii) If in the theory the adjoint matter hypermultiplets are presented,i.e., F pert = F pertV+ F pertA+ f A (m), the WDVV equations never hold.From the investigation of the WDVV equations for the perturbativeprepotentials, one can learn the following lessons:• Masses are to be regarded as moduli.• As an empiric rule, one may say that the WDVV equations are satisfiedby perturbative prepotentials which depend only on the pairwise sums