Energy-momentum tensor of Bosonic Ghost Action in String Theory

Question

When quantizing bosonic string theory by means of the path integral, one inverts the Faddeev-Popov determinant by going to Grassmann variables, yielding:
$$
S_{mathrm{ghosts}} = frac{-i}{2pi}intsqrt{hat{g}}b_{alphabeta}hat{nabla^{alpha}}c^{beta}d^2tau,
$$
where the $-i/2pi$ just comes from convention/Wick rotated or not. My first problem is the notion of the 'fudicial' metric $hat{g}$. I find its role in the path integral procedure a bit confusing. What is its relation to the 'normal' metric $g$? Why is it introduced? Related to this confusion is that fact in my lecture notes it is said that the energy momentum tensor is given by:
$$
T_{alphabeta} :=frac{-1}{sqrt{hat{g}}}frac{delta S_g}{deltahat{g}^{alphabeta}} = frac{i}{4pi}left( b_{alphagamma}hat{nabla}_{beta}c^{gamma} + b_{betagamma}hat{nabla}_{alpha}c^{gamma} - c^{gamma}nabla_{gamma}b_{alphabeta} - g_{alphabeta}b_{gammadelta}nabla^{gamma}c^{delta} right),
$$
I have trouble deriving this. Varying the $sqrt{hat{g}}$ in the action yields the last term I would say:
$$
deltasqrt{hat{g}} = -frac{1}{2}sqrt{hat{g}}hat{g}_{alphabeta}delta hat{g}^{alphabeta}
$$
 However, this term does not have a 'hat' on the covariant derivative, which I find strange. The first and second term follow easily when writing the action with all indices low (except for a factor of 1/2), but I really don't see where the third term comes from and it also misses a hat on the covariant derivative. It looks like there has been done a partial integration, but I don't see why. I guess I am missing the point of the fiducial metric here. Explanation greatly appreciated!

EDIT: In the discussion below I mentioned that $b_{alphabeta}$ is traceless: $b_{alphabeta}g^{alphabeta} = 0$, I forgot to place that here. It is a consequence of the path integral procedure.

Prahar Mitra · Accepted Answer

Here's part of my answer to the derivation of the EM tensor for the ghost action. It does not match the expression you gave, but I may have made a mistake. Can you check my work?
We start with the action
begin{equation}
begin{split}
S_{gh} &= - frac{i}{2pi} int d^2 sigma sqrt{g} g^{alphamu} b_{alphabeta} nabla_mu c^beta 
end{split}
end{equation}
Let us now vary the action w.r.t. metric. We get
begin{equation}
begin{split}
delta S_{gh} &= - frac{i}{2pi} int d^2 sigma left( delta sqrt{g} right) g^{alphamu} b_{alphabeta} nabla_mu c^beta  
&~~~~~~~~~~~~~~~~~- frac{i}{2pi} int d^2 sigma sqrt{g} left( delta g^{alphamu} right) b_{alphabeta} nabla_mu c^beta  
&~~~~~~~~~~~~~~~~~- frac{i}{2pi} int d^2 sigma sqrt{g} g^{alphamu} b_{alphabeta} delta left( nabla_mu c^beta right) 
&= - frac{i}{4pi} int d^2 sigma  sqrt{g} left[ b_{alphamu} nabla_beta c^mu + b_{betamu} nabla_alpha c^mu   - g_{alphabeta}   b_{rhosigma} nabla^rho c^sigma right]  delta  g^{alphabeta}   
&~~~~~~~~~~~~~~~~~ - frac{i}{2pi} int d^2 sigma sqrt{g} g^{alphamu} b_{alphabeta} c^lambda    delta Gamma^beta_{mulambda} 
end{split}
end{equation}
We now use
begin{equation}
begin{split}
delta Gamma^beta_{mulambda} = frac{1}{2} g^{betarho} left[ nabla_lambda delta g_{rho mu} + nabla_mu delta g_{rho lambda} - nabla_rho delta g_{mulambda}right]
end{split}
end{equation}
Note that in particular, it is a tensor. The last term then becomes
begin{equation}
begin{split}
I &=   - frac{i}{2pi} int d^2 sigma sqrt{g} g^{alphamu} b_{alphabeta} c^lambda    delta Gamma^beta_{mulambda}  
&=   - frac{i}{4pi} int d^2 sigma sqrt{g}  b^{murho} c^lambda left[ nabla_lambda delta g_{rho mu} + nabla_mu delta g_{rho lambda} - nabla_rho delta g_{mulambda}right]
&=   - frac{i}{4pi} int d^2 sigma sqrt{g}  b^{murho} c^lambda   nabla_lambda delta g_{rho mu}  
&= - frac{i}{4pi} int d^2 sigma sqrt{g}     nabla_lambda left( b_{alphabeta} c^lambda right)  delta g^{alphabeta} 
end{split}
end{equation}
We then have
begin{equation}
begin{split}
delta S_{gh}  &= - frac{i}{4pi} int d^2 sigma  sqrt{g} left[ b_{alphamu} nabla_beta c^mu + b_{betamu} nabla_alpha c^mu   - g_{alphabeta}   b_{rhosigma} nabla^rho c^sigma +   nabla_lambda left( b_{alphabeta} c^lambda right)  right]  delta  g^{alphabeta}  
end{split}
end{equation}

Energy-momentum tensor of Bosonic Ghost Action in String Theory

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