Polchinsky's Evaluation of the One Loop String Path integral

I try to evaluet the matrix M in the Polchinsky's article(Communications in Mathematical Physics,1986, Volume 104, Issue 1, pp 37-47,"Evaluation of the one loop string path integral",Joseph Polchinski)

(1)$$\det\begin{bmatrix}1 & 0 & 0\\ -D_{a} & \delta_{a}^{c} & 0\\ \frac{1}{2} g^{ef}g_{ef,i} &0 & \delta_{ik}\end{bmatrix} \cdot \det\begin{bmatrix}2+4C & 0 & 0\\ 0 & 2 \Delta_{c}^{d} & -2D_{e} {\chi_{l}^{e}}_{c}\\ 0 &2 \chi_{k}^{ed} D_e & \chi_{kef} \chi_{l}^{ef}\end{bmatrix} \cdot\det\begin{bmatrix}1 & D^{b} & \frac{1}{2} g^{ef}g_{ef,i}\\ 0 & \delta_{d}^{b} & 0\\ 0 & 0 & \delta_{ij}\end{bmatrix}=J N J^{T}$$

where $\Delta_{c}^{d}= -\delta_{c}^{d} D^2-D^{d}D_{c}+D_{c}D^{d} and \chi_{iab}=g_{ab,i}-\frac{1}{2} g_{ab}g^{cd}g_{cd,i}$. I'm considering a torus, so the covariant derivatives become $\partial$. Also the determinant of $J$ and $J^T$ are $1$.

So the determinant of the matrix N become $$\det(N)=\det(2+4C)\det(-2\delta_{c}^{d}\partial^2) \det(\chi_{kef} \chi_{l}^{ef}).$$

This is right?

(2) $\det(\chi_{kef} \chi_{l}^{ef})=\chi_{kef}(g^{e \alpha}g^{f \beta} \chi_{l\alpha \beta})=\partial_k g^{\alpha\beta}\partial_l g_{\alpha \beta}-\frac{1}{2} (g_{\alpha \beta}D_kg^{\alpha \beta})g^{\gamma \delta} \partial_l g_{\gamma \delta}-\frac{1}{2} g^{\alpha \beta} \partial_l g_{\alpha \beta} g^{cd} \partial_k g_{cd}+\frac{1}{2}(g^{cd}\partial_k g^{cd})(g^{\gamma \delta} \partial_k g_{\gamma \delta})$

this is also right?

(3) If steps 1 and 2 are correct, how can I get to the form (given by Polchinski in the article)? $$\det'(N)=\det(2+4C)\det'(-2\delta_{c}^{d} g^{ab} \partial_a \partial_b) \frac{4}{{\tau_2}^4}$$ where the prime denotes the restriction to the zero mode.

• Right, i write wrong...now is correct...thank you! – Valerio Sep 11 '13 at 17:21
• This should be $\delta^{ik}$ (upper indices) in the first matrix, so det($JNJ^T)$ has an indice $^b_{alj}$. So, it does not seem coherent with your expression of det $N$ with indices $^d_{ckl}$ – Trimok Sep 11 '13 at 17:37