# self-adjoint operator $L = *D + D*$ in Witten's work on 3-manifold

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In [Witten Quantum Field Theory and the Jones Polynomial](https://people.maths.ox.ac.uk/beem/papers/jones_polynomial_witten.pdf), 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? It does not even give a uniform differential form in the same dimension in the right hand side?

- 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?

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