# Integration measure for Polyakov's path integral

In section 3.4 of Blumenhagen's Basic Concepts in String Theory, where path integral quatization is presented, and we are given the partition function for the Polyakov's path integral $$Z=\int \mathscr{D} h \mathscr{D} X e^{i S_P[h, X]}\tag{3.63}$$

(I think normally this would be Wick rotated to Euclidian spacetime, which effectively replace the $$i$$ with a minus sign.)

The integration measure is defined as

\begin{aligned} & \|\delta h\|^2=\int d^2 \sigma \sqrt{-h} h^{\alpha \beta} h^{\gamma \delta} \delta h_{\alpha \gamma} \delta h_{\beta \delta}, \\ & \|\delta X\|^2=\int d^2 \sigma \sqrt{-h} \delta X^\mu \delta X_\mu\end{aligned}\tag{3.64} the same as in Polyakov's original paper. But the author states this integration measure is derivable from the following Gaussian integral $$1=\int \mathscr{D}(\delta X) e^{-\frac{1}{4 \pi \alpha^{\prime}}\|\delta X\|^2}\tag{3.65}$$ and other variables are defined "likewise".

I am not sure how $$(3.64)$$ follows from this Gaussian integral definision and I did not find this in Polyakov's paper. Although it is apparent if we make the replacement $$||\delta X||^2\rightarrow h^{\alpha\beta} \partial X^\mu_\alpha X^\nu_\beta \eta_{\mu\nu}$$ we would recover the Polyakov action on the exponent. Also, how would one define the measure for the metric implicitly through a Gaussian integral similar to $$(3.65)$$?

• I sent an email to Prof. Blumehagen for an inquiry about this question, it appears this idea of implicit definition of integration measure comes from Polchinski's 1985 paper Evaluation of the One Loop String Path Integral, but no justification or derivation is presented there, too. Commented Mar 8, 2023 at 21:43