Ivancevic_Applied-Diff-Geom

1106 Applied Differential Geometry: A Modern IntroductionIn order to construct a quantum field theory which will reproduce theSW invariants as correlation functions, anti–ghosts and Lagrangian multipliersare also required. We introduce the anti–ghost multiplet (λ, η)∈ Ω 0 (X), such that[U, λ] = −2λ, [Q, λ] = η, [Q, η] = 0,and the Lagrangian multipliers (χ, H) ∈ Ω 2,+ (X), and (µ, ν) ∈ S − ⊗L suchthat[U, χ] = −χ, [Q, χ] = H, [Q, H] = 0;[U, µ] = −µ, [Q, µ] = ν, [Q, ν] = iφµ.With the given fields, we construct the following functional which hasghost number -1:∫V ={[∇ k ψ k + i (X2 (NM − MN)]λ − χkl H kl − F + kl − i )2 ¯MΓ kl M}− ¯µ (ν − iD A M) − (ν − iD A M)µ , (6.170)where the indices of the tensorial fields are raised and lowered by a givenmetric g on X, and the integration measure is the standard √ gd 4 x. Also,M and ¯µ etc. represent the Hermitian conjugate of the spinorial fields. Ina formal language, M ∈ S + ⊗ L −1 and ¯µ, ¯ν, D A M ∈ S − ⊗ L −1 . Followingthe standard procedure in constructing topological quantum field theory,we take the classical action of our theory to be [Zhang et. al. (1995)]:S = [Q, V ], which has ghost number 0. One can easily show that S isalso BRST invariant, i.e., [Q, S] = 0, thus it is invariant under the fullsuper–algebra (6.169).The bosonic Lagrangian multiplier fields H and ν do not have any dynamics,and so can be eliminated from the action by using their equationsof motionH kl = 1 2(F + kl + i )2 ¯MΓ kl M , ν = 1 2 iD AM. (6.171)Then we arrive at the following expression for the action [Zhang et. al.