Ivancevic_Applied-Diff-Geom

1118 Applied Differential Geometry: A Modern Introductionon the vector field (ψ, N). We also easily get R(λ) = −∆λ + 〈M, M〉λ,where ∆ = d ∗ d. The 〈δν, λ〉 is a two form on A × Γ(W ) whose value on(ψ 1 , N 1 ), (ψ 2 , N 2 ) is -〈N 1 , N 2 〉λ.Combining all the information together, we arrive at the following formula,∫ {π ∗ ρ ∗ (U) = exp − 1 2 |ρ|2 + i(χ, µ)δρ + 2iφµ¯µ+ 〈∆φ, λ〉 − φλ〈M, M〉 + i〈N, N〉λ+ 〈ν, η〉} DχDφDλDηDb. (6.185)Note that the 1–form i(χ, µ)δρ on A × Γ(W ) × Ω 0 (Y ) contacted with thevector field (φ, N, b) leads to2χ k [ −∂ k τ + ∗(∇ψ) k − ¯Mσ k N − ¯Nσ k M ] +2〈µ, [i(D A + b)N − (σ.ψ − τ)M]〉;and the relation (6.184) gives |ρ| 2 = |∗F− ¯MσM| 2 +|db| 2 +|D A M| 2 +b 2 |M| 2 .Finally we get the Euler character∫π ∗ ρ ∗ (U) = exp(−S)DχDφDλDηDb, (6.186)where S is the action (6.178) of the 3D theory defined on the manifold Y .Integrating (6.186) over A × G Γ(W ) leads to the Euler number∑ɛ (A,M) ,[(A,M)]:s(A,M)=0which coincides with the partition function Z of our 3D theory.6.5.3 TQFTs Associated with SW–MonopolesRecall that TQFTs are often used to study topological nature of manifolds.In particular, 3D and 4D TQFTs are well developed. The mostwell–known 3D TQFT would be the Chern–Simons theory, whose partionfunction gives Ray–Singer torsion of 3–manifolds and the other topologicalinvariants can be obtained as gauge invariant observables i.e., Wilsonloops. The correlation functions can be identified with knot or link invariantse.g., Jones polynomal or its generalizations. On the other hand,in 4D, a twisted N = 2 supersymmetric YM theory developed by Witten[Witten (1988a)] also has a nature of TQFT. This YM theory can beinterpreted as Donaldson theory and the correlation functions are identifiedwith Donaldson polynomials, which classify smooth structures of