Let $M$ be a Riemannian manifold and $\sigma$ the world function. The Van-Vleck-Morette determinant $D$ is defined by
$$D(x,x')=\det(-\sigma_{;\mu\nu{}'})$$
Regarding the semi-colon: In chapter $4.1$ of [K] it is claimed that it "denotes differentiation with
respect to the Levi-Civita connection", but in other references it is simply interpreted as partial differentiation$^1$:
$$\sigma_{;\mu\nu{}'}=\frac{\partial^2\sigma}{\partial x^\mu\partial(x')^\nu}$$
In any case, it is not clear to me why $D$ is supposedly coordinate-independent (even if we restrict ourselves to normal coordinates), as it is claimed [here](https://mathoverflow.net/q/135653/281518).

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$^1$

From chapter $4.1$ of [K]:
>[![enter image description here][1]][1]

From page $38$ of [B]:
>[![enter image description here][2]][2]

From [C]:
>[![enter image description here][3]][3]

**References**

[K] Klaus Kirsten, [Spectral Functions in Mathematics and Physics](https://doi.org/10.1201/9781420035469)

[B] Barvinsky and Vilkovisky, [The generalized Schwinger-Dewitt technique in gauge theories and quantum gravity](https://doi.org/10.1016/0370-1573(85)90148-6)

[C] Steven Christensen, [Vacuum expectation value of the stress tensor in an arbitrary curved background: The covariant point-separation method](https://doi.org/10.1103/PhysRevD.14.2490)


  [1]: https://i.sstatic.net/9DWlp.png
  [2]: https://i.sstatic.net/yK5Sdm.png
  [3]: https://i.sstatic.net/zaZeP.png