Adjoint of the differential in Morse-Novikov cohomology Let $M^n$ be a smooth manifold and we can define the Morse-Novikov cohomology group $H^k (M^n,\theta)$ with co-boundary differential operator $d_\theta (w) = dw + \theta \wedge w$  where $0\neq[\theta]∈ H_{dR}^1(M^n)$. This cohomology shares many properties with the ordinary de Rham cohomology and only depend on $[\theta]$.
As it has been shown that the Hodge theory works nicely on this cohomology by defining the adjoint operator $d_\theta^*$ ( I suppose it defines as usual adjoint $\delta$ of de Rham operator $d$ ) and defining the corresponding Laplacian of these operators.
I have the following question: It seems to me that the Stokes’ theorem does not work well with this operator $d_\theta$ because we will have an extra  term which is $\int_M[\theta \wedge w]$and this implies that the Green formula will not work (i.e. $d_\theta^*$ will no longer be adjoint of $d_\theta$). Am I right?
 A: In the paper mentioned in the comments, the operator $d_{\theta}$ is slightly different than the one you mention in your question: $d_{\theta}(w) = dw - \theta\wedge w$. In addition, the operator $\delta_{\theta}$ is defined as $-\ast d_{-\theta}\ast$. In that paper, Otiman is concerned with Inoue surfaces which have real dimension four. In even dimensions, the expression $-\ast d_{-\theta}\ast$ is indeed the adjoint of $d_{\theta}$. In odd dimensions, the adjoint is given by $-\ast d_{(-1)^k\theta}\ast$ when acting on $k$-forms.
To see this, let $\alpha \in \Omega^{k-1}(M)$ and $\beta \in \Omega^k(M)$, then
$$\langle d_{\theta}\alpha, \beta\rangle = \langle d\alpha - \theta\wedge\alpha, \beta\rangle = \langle d\alpha, \beta\rangle - \langle\theta\wedge\alpha, \beta\rangle = \langle\alpha, d^*\beta\rangle - \langle\theta\wedge\alpha, \beta\rangle.$$
Now note that
\begin{align*}
\langle\theta\wedge\alpha, \beta\rangle &= \int_M\theta\wedge\alpha\wedge\ast\beta\\ 
&= \int_M(-1)^{k-1}\alpha\wedge\theta\wedge\ast\beta\\ 
&= \int_M(-1)^{k-1}\alpha\wedge(-1)^{(n-k+1)(k-1)}\ast\ast(\theta\wedge\ast\beta)\\ 
&= (-1)^{(n-k)(k-1)}\int_M\alpha\wedge\ast\ast(\theta\wedge\ast\beta)\\ 
&= (-1)^{n(k-1)}\langle\alpha, \ast(\theta\wedge\ast\beta)\rangle.
\end{align*}
Therefore
\begin{align*}
\langle d_{\theta}\alpha, \beta\rangle &= \langle\alpha, d^*\beta\rangle - \langle\theta\wedge\alpha, \beta\rangle\\ 
&= \langle\alpha, -\ast d\ast\beta\rangle - (-1)^{n(k-1)}\langle\alpha, \ast(\theta\wedge\ast\beta)\rangle\\ 
&= \langle\alpha, -\ast(d\ast\beta + (-1)^{n(k-1)}\theta\wedge\ast\beta)\rangle\\ 
&= \langle\alpha, -\ast(d\ast\beta - (-1)^{n(k-1)+1}\theta\wedge\ast\beta)\rangle\\ 
&= \langle\alpha, -\ast d_{(-1)^{n(k-1)+1}\theta}\ast\beta\rangle
\end{align*}
so the adjoint of $d_{\theta} : \Omega^{k-1}(M) \to \Omega^k(M)$ is $-\ast d_{(-1)^{n(k-1)+1}\theta}\ast$. If $n$ is even, this reduces to $-\ast d_{-\theta}\ast$ (the operator Otiman calls $\delta_{\theta}$), and if $n$ is odd, this reduces to $-\ast d_{(-1)^k\theta}\ast$.
