Ivancevic_Applied-Diff-Geom

762 Applied Differential Geometry: A Modern Introductionequation looks near ɛ = 0 like cɛ − m ¯m = 0, with c a constant. The ɛterm on the left comes from the fact that F + (A 0 ) is proportional to ɛ.Now we can see what happens for ɛ ≠ 0 to the solution that at ɛ = 0 hasA = A 0 , M = M 0 . Depending on the sign of c, there is a solution for m,uniquely determined up to gauge transformation, for ɛ > 0 and no solutionfor ɛ < 0, or vice–versa. Therefore n x jumps by ±1, depending on the signof c, in passing through ɛ = 0.The trivial Abelian instanton with x = 0 is an exception to the abovediscussion, since it cannot be removed by perturbing the metric. To definen 0 , perturb the equation F AB = i 2 (M A ¯M B + M B ¯MA ) toF AB = i 2 (M A ¯M B + M B ¯MA ) − p AB , (4.254)with p a self–dual harmonic 2–form; with this perturbation, the gauge groupacts freely on the solution space. Then define n 0 as the number of gaugeorbits of solutions of the perturbed equations weighted by sign in the usualway. This is invariant under continuous deformations of p for p ≠ 0; aslong as b + 2 > 1, so that the space of possible p’s is connected, the integern 0 defined this way is a topological invariant.The perturbation just pointed out will be used later in the case that p isthe real part of a holomorphic 2–form to compute the invariants of Kählermanifolds with b + 2 > 1. It probably has other applications; for instance,the case that p is a symplectic form is of interest.4.14.10.2 Vanishing TheoremsSome of the main properties of the monopole equations can be understoodby means of vanishing theorems. The general strategy in deriving suchvanishing theorems is quite standard, but some unusual cancellations (requiredby the Lorentz invariance of the underlying untwisted theory) leadto unusually strong results.If we set s = F + − M ¯M, k = DM, then a small calculation gives∫d 4 x √ ( )1gX 2 |s|2 + |k| 2 = (4.255)∫d 4 x √ ( 1g2 |F + | 2 + g ij D i M A D j ¯MA + 1 2 |M|4 + 1 )4 R|M|2 .XHere g is the metric of X, R the scalar curvature, and d 4 x √ g the Riemannianmeasure. A salient feature here is that a term F AB M A ¯M B , which