Relative Invariants, Contact Geometry and Open String Invariants

TL;DR

This paper develops a new theoretical framework for contact and open string invariants, extending relative invariants, by constructing and analyzing specific moduli spaces and proving their compactness and invariance.

Contribution

It introduces two new moduli spaces for contact and open string invariants, establishing their compactness and foundational properties, thus advancing the mathematical understanding of these invariants.

Findings

Defined two new moduli spaces for invariants

Proved compactness of the moduli spaces

Established existence of the contact and open string invariants

Abstract

In this paper we propose a theory of contact invariants and open string invariants, which are generalizations of the relative invariants. We introduce two moduli spaces $\bar{\mathcal{M}}_{A}(M^{+},C,g,m+\nu,{\bf y},{\bf p},(\mathbf{k},\mathfrak{e}))$ and $\bar{\mathcal{M}}_{A}(M,L;g,m+\nu,{\bf y},{\bf p},\overrightarrow{\mu})$, prove the compactness of the moduli spaces and the existence of the invariants.