Let $(M,g)$ and $(N,h)$ be Riemannian manifolds. The action for the nonlinear sigma model with base $M$ and target $N$ is simply the Dirichlet energy (i.e. harmonic map energy) for maps $\phi:M\to N$: $$ S(\phi)=\int_M\|d\phi\|^2\;dV_g. $$ Critical points of $S$ are harmonic maps. For example, when the target $(N,h)$ is just $\mathbb{R}$ with the standard metric, then critical points of $S$ are harmonic functions, i.e. those maps $\phi:M\to \mathbb{R}$ satisfying $\Delta_g\phi=0$.
The partition function for the nonlinear sigma model is formally given by the path integral $$ Z=\int e^{-S(\phi)}\; D\phi. $$
Question: How can we rigorously define $Z$ for general target manifolds $(N,h)$?
Note that there are cases when $Z$ is well defined:
Special case: $(N,h)=(\mathbb{R},g_{\mathrm{std}})$ In this case, we can use zeta function regularization. The Minakshisundaram–Pleijel zeta function $\zeta:\mathbb{C}\to \mathbb{C}$ defined by $$ \zeta(s)=\sum_{\lambda\in\mathrm{Spec}(\Delta_g)\setminus \{0\}} |\lambda|^{-s} \qquad\qquad \text{for}\qquad \mathrm{Re}(s)>\dim M/2 $$ is holomorphic for $\mathrm{Re}(s)>\dim M/2$ and analytically extends to a meromorphic function on all of $\mathbb{C}$ which is regular at $s=0$. We then define the partition function by $$ Z=\det \Delta_g^{-1}=\mathrm{exp}(\zeta'(0)). $$ Can this method be extended to general target manifolds?