# Auxiliary field path integral in non-linear sigma models

I am trying to understand the functional integral over the auxiliary field in the $\mathcal{N}=(2,2)$ supersymmetric non-linear sigma model, or NLSM (reviewed in Chapter 13 of Mirror Symmetry http://www.claymath.org/library/monographs/cmim01c.pdf). There have been previous similar questions on StackExchange, in particular this Phys.SE question: Auxiliary field and loop expansion

Here, the answers seem to indicate that in general, the equations of motion obtained at the clasical level hold at the quantum level, i.e., when one performs the functional integration over the auxiliary field, the result is the same as when one replaces the auxiliary fields by their equation of motion (and drops the corresponding functional integration measure).

However, I have not been able to show this for the NLSM. The auxiliary field functional integral implied by equation 13.4 of Mirror Symmetry is $$Z_F=\int\mathcal{D}F\mathcal{D}\overline{F}\textrm{ }e^{iS_{aux}},$$ where $$S_{aux}=\int d^2x\textrm{ }g_{i\overline{\jmath}}(F^i-\Gamma^i_{jk}\psi_+^j\psi_-^k)(\overline{F}^{\overline{\jmath}}-\Gamma^{\overline{\jmath}}_{\overline{k}\overline{l}}\overline{\psi}_-^\overline{k}\overline{\psi}_+^\overline{l}),$$ (which gives the classical EOM $F^i-\Gamma^i_{jk}\psi_+^j\psi_-^k=0$). This is the analytic continuation of a Gaussian functional integral, and in analogy with the example given here - How does this Gaussian integral over the auxiliary field in 2D topological gauge theory work? , one should be able to perform the functional integral over the auxiliary fields. But due to the presence of the metric, it seems that the result should be $$Z_F\propto \frac{1}{\sqrt{\textrm{det }g}},$$ (based on the classical result $\int_{-\infty}^{\infty}dx\textrm{ }e^{-A_{ij}x^ix^j}=\sqrt{1/\textrm{det }(A/\pi)}$). This is not a constant, as we would expect from the point of the previous paragraph, but rather a function of bosonic fields. Where have I gone wrong?