I am asking this question in order to figure out the expression of the Faddeev-Popov determinant given by Edward Witten is his paper "Quantum Field Theory and Jones Polynomial"
https://projecteuclid.org/euclid.cmp/1104178138
Starting from the action
S[A]=k4π∫MTr(A∧dA+23A∧A∧A)
the variation gives equation of motion
F=dA+A∧A=0
Denoting the solutions to the equation of motion by a, then one expands a generic connection A around a flat connection
A=a+B
Then the action splits into three pieces:
S[A]=k4π∫MTr(a∧da+23a∧a∧a∧a)+k4π∫MTr(B∧DaB)+
+k6π∫MTr(B∧B∧B)
where the covariant derivative in the second term is defined as Da=d+[a,].
The gauge transformation A[U]=U−1dU+U−1AU splits into two parts:
a[U]=U−1dU+U−1aU,B[U]=U−1BU
so that the flat connections remain flat and the perturbation part B lives in the adjoint representation.
Then, the path-integral takes the form
Z=∫DaeiS[a]∫DBexp{ik4π∫MTr(B∧DaB)+ik6π∫MTr(B∧B∧B)}
It's easy to check that the last two terms
S[a;B]=ik4π∫MTr(B∧DaB)+ik6π∫MTr(B∧B∧B)
are indeed invariant under the gauge transformations a[U]=U−1dU+U−1aU and B[U]=U−1BU. Assuming the spacetime manifold M is closed, and k∈Z, then the first part
S[a]==k4π∫MTr(a∧da+23a∧a∧a)
is also gauge invariant up to a shift by 2πZ from the Wess-Zumino-Witten term under large gauge transformations.
For reasons which I still don't understand Assuming that the moduli space of flat connections M=Hom(π1(M),G)/G is a discrete set, then the partition function really is
Z=∑m∈MeiS[am]1Vol∫DBexp{ik4π∫MTr(B∧DaB)+ik6π∫MTr(B∧B∧B)}
To fix the gauge, the author imposes the covariant gauge
F[a;B]=(Da)μBμ=0
The Faddeev-Popov determinant is then
Δ[a;B]=Det(δF[a[U];B[U]]δU)|U=id=Det(M)
Using chain rule, one has
M(x−y)=∫d3z{δF(x)δB(z)δB(z)δU(y)+δF(x)δa(z)δa(z)δU(y)}
In Witten's paper, the final expression of the Faddeev-Popov is quite simple, which is
M=(Da)μ(Da)μ
However, carrying on the calculation of functional derivatives, I obtained a quite different result.
To be specific, the functional derivatives are given by
δF(x)δB(z)=Da(x)δ(x−z),δF(x)δa(z)=[δ(x−z),B(z)]
δB(z)δU(y)=[B(z),δ(z−y)],δa(z)δU(y)=Da(z)δ(z−y)
where (Da)μ(x)δ(x−y)=∂∂xμδ(x−y)+[aμ(x),δ(x−y)], and the commutator here carries Lie-algebra indices and so is symmetric. i.e. [A,B]=AB+BA.
Plugging the above functional derivatives back into the determinant, what I obtained in the end is M(x)=4(Da)μBμ(x)
This is obviously incorrect.
What am I mistaken in the above procedure?