In Witten Quantum Field Theory and the Jones Polynomial, he mentioned,
Let $D$ be the exterior derivative on M, twisted by the fiat connection $A$ let $*$ be the Hodge operator that maps $k$ forms to $3-k$ forms. On a three manifold one has a natural self-adjoint operator $$L = *D + D*$$ which maps differential forms of even order to forms of even order and forms of odd order to forms of odd order. Let $L_-$ denote its restriction to forms of odd order.
In $d$-dimension (say $d=3$),
isn't that $*D$ maps a $k$ form to $d-k-1$-form?
isn't that $D*$ maps a $k$ form to $d-k+1$-form?
So $L = *D + D*$ on $k$-form $V$ produce $LV$ with both $d-k-1$-form and $d-k+1$-form? how can this $L$ operator be natural?
Say $d=3, k=0$, we get $d-k-1=2$-form and $d-k+1=4$-form?
Say $d=3, k=1$, we get $d-k-1=1$-form and $d-k+1=3$-form?
Say $d=3, k=2$, we get $d-k-1=0$-form and $d-k+1=2$-form?
Say $d=3, k=3$, we get $d-k-1=-1$-form and $d-k+1=1$-form?
Letting
\begin{align*} \Omega^{\bullet}(M) &= \bigoplus_{k=0}^{\dim M}\Omega^k(M)\\ \Omega^{\text{even}}(M) &= \bigoplus_{k=0}^{\lfloor\dim M/2\rfloor}\Omega^{2k}(M)\\ \Omega^{\text{odd}}(M) &= \bigoplus_{k=0}^{\lfloor\dim M/2\rfloor}\Omega^{2k+1}(M), \end{align*}
we have $\Omega^{\bullet}(M) = \Omega^{\text{even}}(M)\oplus\Omega^{\text{odd}}(M)$.
The operator $L : \Omega^{\bullet}(M) \to \Omega^{\bullet}(M)$ satisfies $L|_{\Omega^{\text{even}}(M)} : \Omega^{\text{even}}(M) \to \Omega^{\text{even}}(M)$ and $L|_{\Omega^{\text{odd}}(M)} : \Omega^{\text{odd}}(M) \to \Omega^{\text{odd}}(M)$.
In particular, if $\alpha \in \Omega^p(M)$, then $L(\alpha)$ need not be a differential form of pure degree, but it is the sum of such forms whose degrees have the same parity as $p$ does.