Ivancevic_Applied-Diff-Geom

Applied Bundle Geometry 759deformed to the direct sum of the operator d + d ∗ 35 from Λ 1 to Λ 0 ⊕ Λ 2,+and the Dirac operator from S + ⊗L to S − ⊗L. The index of T is the indexof d + d ∗ plus twice what is usually called the index of the Dirac operator;the factor of two comes from looking at S ± ⊗ L as real bundles of twicethe dimension. Let χ and σ be the Euler characteristic and signature ofX. Then the index of d + d ∗ is -(χ + σ)/2, while twice the Dirac indexis -σ/4 + c 1 (L) 2 . The virtual dimension of the moduli space is the sum ofthese or2χ + 3σW = − + c 1 (L) 2 .4When this number is negative, there are generically no solutions of themonopole equations. When W = 0, that is, when x = −c 1 (L 2 ) = −2c 1 (L)obeysx 2 = 2χ + 3σ, (4.252)then the virtual dimension is zero and the moduli space generically consistsof a finite set of points P i,x , i = 1 . . . t x . With each such point, one canassociate a sign ɛ i,x = ±1 – the sign of the determinant of T as we discussmomentarily. Once this is done, define for each x obeying (4.252) an integern x byn x = ∑ iɛ i,x .We will see later that n x = 0 – indeed, the moduli space is empty – for allbut finitely many x. Under certain conditions that we discuss in a moment,the n x are topological invariants.Note that W = 0 iff the index of the Dirac operator is∆ = χ + σ .4In particular, ∆ must be an integer to have non–trivial n x . Similarly, ∆must be integral for the Donaldson invariants to be non–trivial (otherwiseSU(2) instanton moduli space is odd–dimensional).For the sign of the determinant of T to make sense one must trivializethe determinant line of T . This can be done by deforming T as above tothe direct sum of d + d ∗ and the Dirac operator. If the Dirac operator,which naturally has a non–trivial complex determinant line, is regarded asa real operator, then its determinant line is naturally trivial – as a complex35 What is meant here is a projection of the d + d ∗ operator to self–dual forms.