Ivancevic_Applied-Diff-Geom

Applied Bundle Geometry 727pear at the trivial section ψ ≡ 0, since elsewhere the action of the gaugegroup is free. Hence, by perturbing the equation, it is possible to get asmooth moduli space.The analogue of the vanishing Theorem for Donaldson polynomials ona manifold that splits as a connected sum can be proven, reinforcing theintuitive feeling that the two sets of invariants ought to be the same.Moreover, explicit computations can be done in the case of Kähler manifolds,by looking at the SW invariants associated to the canonical linebundle.The latter result has a generalization due to [Taubes (1994)], where itis shown that the value ±1 of the SW invariants is achieved on symplecticfour–manifolds, with respect to the canonical line bundle, by a techniquethat involves estimates of solutions of a parametrized family of perturbedmonopole equations.4.14.2.1 Clifford Algebras and Dirac OperatorsRecall that the Clifford algebra C(V ) of a (real or complex) vector space Vwith a symmetric bilinear form (, ) is the algebra generated by the elements[Marcolli (1995)] {e ɛ11 · · · eɛn n , }, where ɛ i = 0, or 1 and {e i } is an orthogonalbasis of V , subject to the relations e·e ′ +e ′·e = −2(e, e ′ ). The multiplicationof elements of V in the Clifford algebra is called Clifford multiplication.In particular given a differentiable manifold X we shall consider theClifford algebra associated to the tangent space at each point. The Cliffordalgebra of the tangent bundle of X is the bundle that has fibre over eachpoint x ∈ X the Clifford algebra C(T x X). We shall denote this bundleC(T X).If dim V = 2m, there is a unique irreducible representation of the Cliffordalgebra C(V ). This representation has dimension 2 m .A spinor bundle over a Riemannian manifold X is the vector bundleassociated to C(T X) via this irreducible representation, endowed with aHermitian structure such that the Clifford multiplication is skew–symmetricand compatible with the Levi–Civita connection on X (see [Roe (1988)]).Not all manifolds admit a spinor bundle; it has been proved in [Roe(1988)] that the existence of such a bundle is equivalent to the existence ofa Spin c –structure on the manifold X: we shall discuss Spin c −-structuresin the next paragraph. If such a bundle exists, it splits as a direct sumof two vector bundles, S = S + ⊕ S − , where the splitting is given by theinternal grading of the Clifford algebra.