Choose a reference metric in each conformal class. An infinitesimal worldsheet diffeomorphism generated by

c^{α}

and a Weyl transformation decompose the metric variation into its trace and the traceless operator

(P_{1} c)_{α β} = \nabla_{α} c_{β} + \nabla_{β} c_{α} - g_{α β} \nabla_{γ} c^{γ} .

(1)

Inserting the gauge condition and its Faddeev-Popov determinant cancels the formal gauge-orbit volume. Representing

det P_{1}

by anticommuting worldsheet ghost fields gives

S_{Gh} [b, c, g] = \frac{1}{2 π} \int_{Σ} d^{2} σ g b^{α β} (P_{1} c)_{α β},

(2)

where

b_{α β}

is symmetric and traceless and

c^{α}

is a vector. The gauge-fixed integral is consequently

Z = \int D X D b D c exp (- S_{P} - S_{Gh}) .

(3)

On higher-genus worldsheets one must additionally integrate over moduli and treat conformal-Killing and ghost zero modes separately; these finite-dimensional factors are suppressed in the displayed formal expression.

Solved by gpt-5.6-sol high.

b

 0  0

Solution

 0  0

In Euclidean conformal gauge, complex coordinates reduce the worldsheet ghost action to

S_{Gh} = \frac{1}{2 π} \int d^{2} z (b_{zz} \overset{ˉ}{\partial} c^{z} + b_{\overset{z}{ˉ} \overset{z}{ˉ}} \partial c^{\overset{z}{ˉ}})

(1)

up to a convention-dependent common normalization absorbed into the fields. Inverting

\overset{ˉ}{\partial}

and

\partial

gives

⟨ b_{zz} (z) c^{z} (w)⟩ = \frac{1}{z - w}, ⟨ b_{\overset{z}{ˉ} \overset{z}{ˉ}} (\overset{z}{ˉ}) c^{\overset{z}{ˉ}} (\overset{w}{ˉ})⟩ = \frac{1}{z ˉ - w ˉ},

(2)

with mixed holomorphic-antiholomorphic correlators equal to zero. These are the standard local-plane propagators; compact worldsheets require the zero-mode qualifications described in part (a).

Solved by gpt-5.6-sol high.