Definition of exterior derivative from a connection

Question

I fail to see what is the meaning of the symbol $d_{\nabla}$ in (1.2) of

http://arxiv.org/pdf/hep-th/9712042v2.pdf

I know the meaning of that symbol in the context of forms taking values on some vector bundle with connection $\nabla$, but this is different since it is a boundary operator of the standard de Rham complex which at degree zero acts as $\nabla$.

Thanks.

Answer 1

I will assume that you know about connections on the tangent bundle. These connections induce connections on the tensor bundle $\mathcal{T}^{r,s}M$ of $(r,s)$ tensor fields. Given $∇ : Γ(TM) \to Γ(TM \otimes T^{*}M)$ there is a unique connection $d_∇ : Γ(\mathcal{T}^{r,s}M) \to Γ(\mathcal{T}^{r,s}M \otimes T^{*}M)$ satisfying

• $d_∇ = ∇$ on $TM$,
• $(d_∇X)f = X(f)$ for functions $f ∈ C^{∞}(M)$,
• Product rule: $(d_∇X)(F \otimes G) = d_∇X F \otimes G + F \otimes d_∇X G$,
• Trace invariance: $d_∇X(\mathrm{tr}(F)) = \mathrm{tr}(d_∇X F)$.

The almost complex structure $J ∈ Γ(TM \otimes T^{*}M)$ is a $(1,1)$ tensor. Similarly, this works for forms.