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?



Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.