Ivancevic_Applied-Diff-Geom

Applied Bundle Geometry 753equating a and a D with periods of a certain meromorphic 1–form λ on thecurve E. λ has two characteristics:(i) λ may have poles but (as long as the monodromies are in SL(2, Z))its residues vanish; and(ii) To achieve positivity of the metric on the quantum moduli space,its derivative with respect to u is proportional to dx y .Condition (i) means that the definition of a and a D by contour integrals∮∮a = λ, a D = λ,γ 1 γ 2(with γ 1 and γ 2 some contours on E) is invariant under deformation of theγ i , even across poles of λ. This ensures that only the homology classes ofthe γ i matter and reduces the monodromies to a group SL(2, Z) that actson H 1 (E, Z). In the presence of bare masses, this is too strong a conditionsince when the bare masses are non–zero the monodromies are not quite inSL(2, Z).As for condition (ii), the differential form dx yhas no poles and representsa cohomology class on E of type (1, 0). Having dλ/du = f(u) dx/y leadsto positivity of the metric. The function f(u) is determined by requiringthe right behavior at the singularities, for instance a ≈ 2√ 1 2u for large u,while f is a constant. The proper relation is in fact√dλ 2du = 8πdxy . (4.245)Up to an inessential sign, this is 1/2 the value in the ‘old’ conventions.By integration with respect to u, (4.245) determines λ (once the curve isknown) for all values of N f . This relation is only supposed to hold up to atotal differential in x; λ is supposed to be meromorphic in x.The massless N f = 3 curve is given byy 2 = x 2 (x − u) − (x − u) 2 .The polynomial on the right hand side has zeroes at x 0 = u and atx ± = 1 2(1 ±√1 − 4u).In particular, at u = 1/4, x + = x − , giving the singularity that we haveattributed to a massless state of (n m , n e ) = (2, 1). To show that this stateis semiclassical, we interpolate on the positive u axis from the semiclassical