This is a question about notation, I apologize if it is too basic. In the paper
Cho, Koji; Matsumoto, Keiji, Intersection theory for twisted cohomologies and twisted Riemann’s period relations. I, Nagoya Math. J. 139, 67-86 (1995). ZBL0856.32015.
certain interesting identities for beta function are derived via twisted cohomology theory. I struggle to understand the basic notation. In the introduction, a certain logarithmic 1-form $\omega$ on $\mathbf{P}^1$ is introduced, and then two spaces
$$ \Gamma(\mathbf{P}^1, \Omega^1(\log D))/\mathbf{C}\cdot \omega,\quad \Gamma(\mathbf{P}^1, \Omega^1(\log D))/\mathbf{C}\cdot (-\omega). $$ are claimed to be isomorphic to the first cohomology group of local systems defined by connections $\nabla =d+\omega\wedge $ and $\nabla = d-\omega\wedge $, respectively. But if this notation denotes what I presume it should denote - the quotient of the space of global sections of the sheaf of logarithmic 1-forms by the one-dimentional space generated by $\omega$ - then these two spaces are just the same space! However, the authors clearly distinguish between them throughout the text.
So, the question is: what does the notation mean? If these are indeed the same space, what do the authors try to convey by denoting them differently?
