Pseudoholomoprhic curves on the $\mathfrak{LCS}$-fication of contact manifolds

Abstract: For each contact diffeomorphism $\phi: (Q,\xi) \to (Q,\xi)$ of $(Q,\xi)$, we equip its mapping torus $M_\phi$ with a \emph{locally conformal symplectic} form of Banyaga's type, which we call the \emph{$\text{\rm lcs}$ mapping torus} of contact diffeomorphism $\phi$. In the present paper, we consider the product $Q \times S^1= M_{id}$ (corresponding to $\phi = id$) and develop basic analysis of the associated $J$-holomorphic curve equation, which has the form $$ \overline{\partial}^\pi w = 0, \quad w^*\lambda\circ j = f^*d\theta $$ for the map $u = (w,f): \dot \Sigma \to Q \times S^1$ for the $\lambda$-compatible almost complex structure $J$ and a punctured Riemann surface $(\dot \Sigma, j)$. In particular, $w$ is a \emph{contact instanton} in the sense of [OW2, OW3]. We develop a scheme of treating the non-vanishing charge by introducing the notion of \emph{charge class} in $H^1(\dot \Sigma,\mathbb Z)$ and develop the geometric framework for the study of pseudoholomorphic curves, a correct choice of energy and the definition of moduli spaces towards the construction of compactification of the moduli space on the $\mathfrak{lcs}$-fication of $(Q,\lambda)$ (more generally on arbitrary locally conformal symplectic manifolds).