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

TL;DR

This paper develops a framework for studying pseudoholomorphic curves on the locally conformal symplectic (lcs) mapping torus of contact manifolds, introducing contact instantons, charge classes, and moduli space compactifications.

Contribution

It introduces a geometric framework for analyzing pseudoholomorphic curves on lcs manifolds, including charge classes and moduli space construction, extending contact instanton theory.

Findings

Defined charge class in $H^1( ext{punctured surface}, \\mathbb{Z})$

Developed energy and moduli space concepts for lcs-fication

Extended contact instanton analysis to lcs 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).