Ivancevic_Applied-Diff-Geom

1104 Applied Differential Geometry: A Modern Introductionwhere D A is the twisted Dirac operator, Γ ij = 1 2 [γ i, γ j ], and F + representsthe self–dual part of the curvature of L with connection A.If X is not a spin manifold, then spin bundles do not exist. However,it is always possible to introduce the so called Spin c bundles S ± ⊗ L,with L 2 being a line bundle. Then in this more general setting, the SWmonopoles equations look formally the same as (6.167), but the M shouldbe interpreted as a section of the the Spin C bundle S + ⊗ L.Denote by M the moduli space of solutions of the SW monopole equationsup to gauge transformations. Generically, this space is a manifold.Its virtual dimension is equal to the number of solutions of the followingequations(dψ) + kl + i 2( ¯MΓkl N + ¯NΓ kl M ) = 0, D A N + ψM = 0,∇ k ψ k + i (NM − MN) = 0, (6.168)2where A and M are a given solution of (6.167), ψ ∈ Ω 1 (X) is a oneform, (dψ) + ∈ Ω 2,+ (X) is the self dual part of the two form dψ, andN ∈ S + ⊗ L. The first two of the equations in (6.168) are the linearizationof the monopole equations (6.167), while the last one is a gauge fixingcondition. Though with a rather unusual form, it arises naturally from thedual operator governing gauge transformationsLetC : Ω 0 (X) → Ω 1 (X) ⊕ (S + ⊗ L),φ ↦→ (−dφ, iφM).T : Ω 1 (X) ⊕ (S + ⊗ L) → Ω 0 (X) ⊕ Ω 2,+ (X) ⊕ (S − ⊗ L),be the operator governing equation (6.168), namely, the operator whichallows us to rewrite (6.168) as T (ψ, N) = 0. Then T is an elliptic operator,the index Ind(T ) of which yields the virtual dimension of M. Astraightforward application of the Atiyah–Singer index Theorem gives2χ(X) + 3σ(X)Ind(T ) = − + c 1 (L) 2 ,4where χ(X) is the Euler character of X, σ(X) its signature index and c 1 (L) 2is the square of the first Chern class of L evaluated on X in the standardway.When Ind(T ) equals zero, the moduli space generically consists of afinite number of points, M = {p t : t = 1, 2, ..., I}. Let ɛ t denote the signof the determinant of the operator T at p t , which can be defined with