I met a problem of derving the weyl transformation on the closed-string tachyon, Eq. (3.6.8) in Polchinski's string theory, vol 1, p 103.

Given the vertax operator of the closed-string tachyon

$$V_0 = 2g_c \int d^2 \sigma g^{1/2} e^{ik \cdot X} \tag{3.6.1}$$

It is said the Weyl transformation of Eq. (3.6.1) is

$$\delta_W V_0 = 2g_c \int d^2 \sigma g^{1/2} \left( 2 \delta \omega( \sigma) - \frac{k^2}{2} \delta_W \Delta (\sigma, \sigma) \right) [ e^{ik \cdot X(\sigma)} ]_r \tag{3.6.8}$$

here renormalized operator $[]_r$ and $\Delta (\sigma, \sigma)$ are defined as

$$ [ \mathcal{F}]_r = \exp \left( \frac{1}{2} \int d^2 \sigma d^2 \sigma' \Delta(\sigma, \sigma') \frac{ \delta}{\delta X^{\mu}(\sigma)} \frac{ \delta}{\delta X_{\mu}(\sigma')} \right) \mathcal{F} \tag{3.6.5} $$ $$ \Delta(\sigma,\sigma') = \frac{ \alpha'}{2} \ln d^2(\sigma,\sigma') \tag{3.6.6} $$ where $d(\sigma,\sigma')$ is the geodesic distance between points $\sigma$ and $\sigma'$.

I can get the first term in the big parathesis $()$ of Eq. (3.6.8), namely consider $$ \delta g^{1/2} = \frac{1}{2} \frac{1}{g^{1/2}} \delta g= \frac{1}{2} \frac{1}{g^{1/2}} g g^{ab} \delta g_{ab} = \frac{1}{2} \frac{1}{g^{1/2}} g g^{ab} 2 \delta \omega g_{ab} = 2 g^{1/2} \delta \omega $$ I try to get the second term in the big parathesis of Eq. (3.6.8) $$ \delta e^{i k \cdot X} = e^{i k \cdot X} i k^{ab} X^{ab} \delta g_{ab} = e^{i k \cdot X} i k^{ab} X^{ab} 2 \delta \omega g_{ab} = 2 i k \cdot X e^{ik \cdot X} \delta \omega,$$

but where do the $\Delta(\sigma,\sigma)$ and renormalized opeator $ [ e^{ik \cdot X(\sigma)} ]_r $ in (3.6.8) come from?

I met a problem of derving the Weyl transformation on the closed-string tachyon, Eq. (3.6.8) in Polchinski's string theory, vol 1, p 103.

Given the vertax operator of the closed-string tachyon

$$V_0 = 2g_c \int d^2 \sigma g^{1/2} e^{ik \cdot X} \tag{3.6.1}$$

It is said the Weyl transformation of Eq. (3.6.1) is

$$\delta_W V_0 = 2g_c \int d^2 \sigma g^{1/2} \left( 2 \delta \omega( \sigma) - \frac{k^2}{2} \delta_W \Delta (\sigma, \sigma) \right) [ e^{ik \cdot X(\sigma)} ]_r \tag{3.6.8}$$

here renormalized operator $[]_r$ and $\Delta (\sigma, \sigma)$ are defined as

$$ [ \mathcal{F}]_r = \exp \left( \frac{1}{2} \int d^2 \sigma d^2 \sigma' \Delta(\sigma, \sigma') \frac{ \delta}{\delta X^{\mu}(\sigma)} \frac{ \delta}{\delta X_{\mu}(\sigma')} \right) \mathcal{F} \tag{3.6.5} $$ $$ \Delta(\sigma,\sigma') = \frac{ \alpha'}{2} \ln d^2(\sigma,\sigma') \tag{3.6.6} $$ where $d(\sigma,\sigma')$ is the geodesic distance between points $\sigma$ and $\sigma'$.

I can get the first term in the big parathesis $()$ of Eq. (3.6.8), namely consider $$ \delta g^{1/2} = \frac{1}{2} \frac{1}{g^{1/2}} \delta g= \frac{1}{2} \frac{1}{g^{1/2}} g g^{ab} \delta g_{ab} = \frac{1}{2} \frac{1}{g^{1/2}} g g^{ab} 2 \delta \omega g_{ab} = 2 g^{1/2} \delta \omega $$ I try to get the second term in the big parathesis of Eq. (3.6.8) $$ \delta e^{i k \cdot X} = e^{i k \cdot X} i k^{ab} X^{ab} \delta g_{ab} = e^{i k \cdot X} i k^{ab} X^{ab} 2 \delta \omega g_{ab} = 2 i k \cdot X e^{ik \cdot X} \delta \omega,$$

but where do the $\Delta(\sigma,\sigma)$ and renormalized opeator $ [ e^{ik \cdot X(\sigma)} ]_r $ in (3.6.8) come from?

I met a problem of derving the weyl transformation on the closed-string tachyon, Eq. (3.6.8) in Polchinski's string theory, vol 1, p 103.

Given the vertax operator of the closed-string tachyon

$$V_0 = 2g_c \int d^2 \sigma g^{1/2} e^{ik \cdot X} \tag{3.6.1}$$

It is said the Weyl transformation of Eq. (3.6.1) is

$$\delta_W V_0 = 2g_c \int d^2 \sigma g^{1/2} \left( 2 \delta \omega( \sigma) - \frac{k^2}{2} \delta_W \Delta (\sigma, \sigma) \right) [ e^{ik \cdot X(\sigma)} ]_r \tag{3.6.8}$$

here renormalized operator $[]_r$ and $\Delta (\sigma, \sigma)$ are defined as

$$ [ \mathcal{F}]_r = \exp \left( \frac{1}{2} \int d^2 \sigma d^2 \sigma' \Delta(\sigma, \sigma') \frac{ \delta}{\delta X^{\mu}(\sigma)} \frac{ \delta}{\delta X_{\mu}(\sigma')} \right) \mathcal{F} \tag{3.6.5} $$ $$ \Delta(\sigma,\sigma') = \frac{ \alpha'}{2} \ln d^2(\sigma,\sigma') \tag{3.6.6} $$

I can get the first term in the big parathesis $()$ of Eq. (3.6.8), namely consider $$ \delta g^{1/2} = \frac{1}{2} \frac{1}{g^{1/2}} \delta g= \frac{1}{2} \frac{1}{g^{1/2}} g g^{ab} \delta g_{ab} = \frac{1}{2} \frac{1}{g^{1/2}} g g^{ab} 2 \delta \omega g_{ab} = 2 g^{1/2} \delta \omega $$ I try to get the second term in the big parathesis of Eq. (3.6.8) $$ \delta e^{i k X} = e^{i k X} i k^{ab} X^{ab} \delta g_{ab} = e^{i k X} i k^{ab} X^{ab} 2 \delta \omega g_{ab} = 2 i k X e^{ikX} \delta \omega,$$

but where do the $\Delta(\sigma,\sigma)$ and renormalized opeator $ [ e^{ik \cdot X(\sigma)} ]_r $ in (3.6.8) come from?

I met a problem of derving the weyl transformation on the closed-string tachyon, Eq. (3.6.8) in Polchinski's string theory, vol 1, p 103.

Given the vertax operator of the closed-string tachyon

$$V_0 = 2g_c \int d^2 \sigma g^{1/2} e^{ik \cdot X} \tag{3.6.1}$$

It is said the Weyl transformation of Eq. (3.6.1) is

$$\delta_W V_0 = 2g_c \int d^2 \sigma g^{1/2} \left( 2 \delta \omega( \sigma) - \frac{k^2}{2} \delta_W \Delta (\sigma, \sigma) \right) [ e^{ik \cdot X(\sigma)} ]_r \tag{3.6.8}$$

here renormalized operator $[]_r$ and $\Delta (\sigma, \sigma)$ are defined as

$$ [ \mathcal{F}]_r = \exp \left( \frac{1}{2} \int d^2 \sigma d^2 \sigma' \Delta(\sigma, \sigma') \frac{ \delta}{\delta X^{\mu}(\sigma)} \frac{ \delta}{\delta X_{\mu}(\sigma')} \right) \mathcal{F} \tag{3.6.5} $$ $$ \Delta(\sigma,\sigma') = \frac{ \alpha'}{2} \ln d^2(\sigma,\sigma') \tag{3.6.6} $$ where $d(\sigma,\sigma')$ is the geodesic distance between points $\sigma$ and $\sigma'$.

I can get the first term in the big parathesis $()$ of Eq. (3.6.8), namely consider $$ \delta g^{1/2} = \frac{1}{2} \frac{1}{g^{1/2}} \delta g= \frac{1}{2} \frac{1}{g^{1/2}} g g^{ab} \delta g_{ab} = \frac{1}{2} \frac{1}{g^{1/2}} g g^{ab} 2 \delta \omega g_{ab} = 2 g^{1/2} \delta \omega $$ I try to get the second term in the big parathesis of Eq. (3.6.8) $$ \delta e^{i k \cdot X} = e^{i k \cdot X} i k^{ab} X^{ab} \delta g_{ab} = e^{i k \cdot X} i k^{ab} X^{ab} 2 \delta \omega g_{ab} = 2 i k \cdot X e^{ik \cdot X} \delta \omega,$$

but where do the $\Delta(\sigma,\sigma)$ and renormalized opeator $ [ e^{ik \cdot X(\sigma)} ]_r $ in (3.6.8) come from?