Ivancevic_Applied-Diff-Geom

Applied Bundle Geometry 781where we have set ω 1 = −iπ without loss of generality. The series expansion(4.286) is superconvergent and sparse in the sense that it receivescontributions only at integers that grow like n 2 .2. The prepotential of the SW theory obeys a renormalization group–type equation that simply relates F to the CM Hamiltonian, expressed interms of the quantum order parameters a ja j = 1 ∮dλ,2πi A j∂F∂τ | a j= H(x, p) = 1 2 TrL(z)2 + C℘(z). (4.287)Furthermore, in an expansion in powers of the instanton factor q = e 2πiτ ,the quantum order parameters a j may be computed by residue methods interms of the zeros of H(k). The proof of (4.287) requires Riemann surfacedeformation theory. The fact that the quantum order parameters may beevaluated by residue methods arises from the fact that A j −cycles may bechosen on the spectral curve Γ in such a way that they will shrink to zeroas q → 0. As a result, contour integrals around full-fledged branch cutsA j reduce to contour integrals around poles at single points, which may becalculated by residue methods only. Knowing the quantum order parametersin terms of the zeros k j of H(k) = 0 is a relation that may be invertedand used in (4.287) to get a differential relation for all order instanton corrections.It is now only necessary to evaluate explicitly the τ−independentcontribution to F, which in field theory arises from perturbation theory.This may be done easily by retaining only the n = 0 and n = 1 terms inthe expansion of the curve (4.286), so that z = ln H(k) − ln H(k − m). Theresults of the calculations to two instanton order may be summarized inthe following Theorem:3. The prepotential, to 2 instanton order is given by F = F pert +F (1) +F (2) . The perturbative contribution is given byF pert = τ 2∑ia 2 i − 18πi∑ [(ai − a j ) 2 ln(a i − a j ) 2 − (a i − a j − m) 2 ln(a i − a j − m) 2i,j(4.288)while all instanton corrections are expressed in terms of a single functionS i (a) =∏ Nj=1 [(a i − a j ) 2 − m 2 ]∏j≠i (a − a j) 2 , as follows: