I am studying nonlinear sigma-models and topological twists using E.Witten's article "Mirror manifolds and topological field theory" (https://arxiv.org/abs/hep-th/9112056), as well as "Mirror symmetry" book by Hori, Katz, etc, and I am trying to understand how NLSM's on curved worldsheets work. I am having trouble with why is the Lagrangian of the model well defined, that is, invariant under diffeomorphisms of worldsheet.
Considering the maps $\Phi:\Sigma \to X$ for Kahler manifold $X$, the following Lagrangian for SUSY ${\cal N}=(2,2)$ NLSM is studied by Witten: $$ L=\text{(bosonic term) +(4-fermion term)}+ i \psi^\overline{i}_-D_z \psi^i_- g_{i \overline{i}} + i \psi^\overline{i}_+ D_\overline{z} \psi^i_+ g_{i \overline{i}}. $$ Here fermion fields are sections of spinor bundle on $\Sigma$ with values in $\Phi^*TX$. The fact that they are in a pullback bundle is accounted for by using covariant derivative $D_z$ for fermions instead of ordinary $\partial_z$ one; it includes Levi-Civita connection on $X$ $$ D_z \psi^i = \partial_z \psi^i + \partial_z\phi^j \Gamma^i_{jk} \psi^k $$ so the Lagrangian behaves well under the action of diffeomorphisms on $X$.
Yet, the worldsheet $\Sigma$ is supposed to be a Riemann surface and in general is also not flat. The thing I don't get is why then we don't write spin connection here as well, if we consider an object from spinor bundle? If we don't do that, then invariance under diffeomorphisms of $\Sigma$ doesn't seem to hold.
I was able to check the invariance of a given action under proposed SUSY transformations; but it looks for me like including spin connection would ruin it. Am I missing something important here? Is something silently implied when the following Lagrangian is written?
